Characterizing Bipartite Distance-Regularized Graphs with Vertices of Eccentricity 4

Abstract

Consider a bipartite distance-regularized graph $\Gamma$ with color partitions Y and $Y'$. Notably, all vertices in partition Y (and similarly in $Y'$) exhibit a shared eccentricity denoted as D (and $D'$, respectively). The characterization of bipartite distance-regularized graphs, specifically those with $D \le 3$, in relation to the incidence structures they represent is well established. However, when $D=4$, there are only two possible scenarios: either $D'=3$ or $D'=4$. The instance where $D=4$ and $D'=3$ has been previously investigated. In this paper, we establish a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v, b, r, k, \lambda _1, 0)$ of type $(k-1, t)$, featuring intersection numbers $x=0$ and $y>0$ (where $y \le t < k$), and bipartite distance-regularized graphs with $D=D'=4$. Moreover, our investigations result in the systematic classification of 2-Y-homogeneous bipartite distance-regularized graphs, which are incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$.

1 Introduction

The characterization of bipartite distance-regularized graphs with some vertices of eccentricity less than 4, in terms of the incidence structures of which they are incidence graphs, is complete (see [1, 9, 14]). It is known by the result of Godsil and Shawe-Taylor that every distance-regularized graph is either distance-regular or distance-biregular (see [10]). Bipartite distance-regular graphs with diameter 3 are exactly the incidence graphs of symmetric 2-designs and are well studied (see [1, 14]). Moreover, there is a one-to-one correspondence between distance-biregular graphs with vertices of eccentricity three and four (depending on the part) and the incidence graphs of quasi-symmetric 2-designs with one intersection number zero (see [9, Theorem 5.4]).

In this paper, we consider bipartite distance-regularized graphs with all vertices of eccentricity 4. The outcomes detailed herein can be seen as a contribution to the characterization of bipartite distance-regularized graphs in terms of the incidence structures of which they are incidence graphs. The main result is given in Theorem 6.4, where we prove that bipartite distance-regularized graphs with all vertices of eccentricity 4 are exactly the incidence graphs of quasi-symmetric special partially balanced incomplete block designs with parameters $(v,b,r,k, \lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y> 0$, where $y\le t < k$.

The property of being (almost) 2-Y-homogeneous turned out to be a desirable property of a bipartite distance-regularized graph. As such, it was studied by Curtin and Nomura for bipartite distance-regular graphs (see [3,4,5, 12]). Recently, Penjić and the first author studied the intersection arrays of distance-biregular graphs and gave sufficient and necessary conditions under which such graphs are (almost) 2-Y-homogeneous (see [8]). Furthermore, the 2-Y-homogeneous condition within the incidence graphs of 2-designs was studied in detail by the first and third authors (see [9]). However, the comprehensive classification of all graphs with this property remains an open question. This motivates us to study the 2-Y-homogeneous condition of a bipartite distance-regularized graph where all vertices have eccentricity 4. Our investigations lead to a classification of 2-Y-homogeneous bipartite distance-regularized graphs which are incidence graphs of quasi-symmetric special partially balanced incomplete block designs with parameters $(v,b,r,k, \lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$, where $y\le t <k$.

The paper is structured as follows. In the next section, we provide the relevant background information. In Sect. 3, we give some properties of bipartite distance-regularized graphs, including the (almost) 2-Y-homogeneous condition of bipartite distance-regularized graphs. In Sect. 4, we introduce the concept of special partially balanced incomplete block designs. Then, in Sect. 5, we relate bipartite distance-regularized graphs with incidence graphs of special partially balanced incomplete block designs. Finally, in Sect. 6 we consider distance-semiregular graphs and special partially balanced incomplete block designs. We substantiate our principal result, establishing a precise one-to-one correspondence between the incidence graphs of quasi-symmetric special partially balanced incomplete block designs with parameters $(v,b,r,k, \lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $y\le t<k$, and bipartite distance-regularized graphs with $D=D'=4$ (Theorem 6.4). We also classify 2-Y-homogeneous bipartite distance-regularized graphs, which are incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$ (Theorem 6.5).

2 Preliminaries

We assume familiarity with the basic facts and notions from graph theory and from the theory of combinatorial designs. For background reading, we refer the reader to [1, 2, 7, 13].

In this paper, $\Gamma =(X, \mathcal {R})$ will denote a finite, undirected, connected graph, without loops and multiple edges, with vertex set X and edge set $\mathcal {R}$. An incidence structure $\mathcal {D} =( \mathcal {P}, \mathcal {B}, I)$, with point set $\mathcal {P}$, block set $\mathcal {B}$ and incidence relation $I \subseteq \mathcal {P} \times \mathcal {B}$, where $|\mathcal {P}|=v$, $|\mathcal {B}|=b$, each block $B \in \mathcal {B}$ is incident with exactly k points, every t-tuple of distinct points from $\mathcal {P}$ is incident with exactly $\lambda$ blocks, and each point incident with exactly r blocks is a t-$(v,b,r,k,\lambda )$ design or a t-$(v,k,\lambda )$ design. We will only consider t-$(v, k, \lambda )$ designs that are simple, proper and nontrivial, and to rule out degenerate cases, we will assume that the parameters of a design satisfy $v>k>t\ge 1$ and $\lambda \ge 1$.

Consider a graph $\Gamma =(X, \mathcal {R})$, and, for any $x, y\in X$, denote by $\partial (x, y)$ the distance between x and y (the length of a shortest walk from x to y). The diameter of $\Gamma$ is defined to be $\max \{\partial (u,v)\,|\,u, v\in X\},$ and the eccentricity of x, denoted by $\varepsilon =\varepsilon (x)$, is the maximum distance between x and any other vertex of $\Gamma$. Note that the diameter of $\Gamma$ equals $\max \{\varepsilon (x)\mid x\in X\}$. For an integer i, we define $\Gamma _i(x)=\left\{ y \in X \mid \partial (x, y)=i\right\} .$ Notice that $\Gamma _{i}(x)$ is empty if and only if $i<0$ or $i>\varepsilon (x)$, and $\Gamma _1(x)$ is the set of neighbors of x. We will abbreviate $\Gamma (x)=\Gamma _1(x)$. We say that a vertex $x\in X$ has valency k if $|\Gamma (x)|=k$. A graph $\Gamma$ is called regular if every vertex has the same valency, i.e., if there is a nonnegative integer k such that $|\Gamma (x)|=k$ for every vertex $x\in X$. In this case, we also say that $\Gamma$ is regular with valency k or k-regular.

A bipartite (or $(Y,Y')$-bipartite) graph is a graph whose vertex set can be partitioned into two subsets Y and $Y'$ such that each edge has one end in Y and one end in $Y'$. The vertex sets Y and $Y'$ in such a partition are called color partitions (or bipartitions) of the graph. A bipartite graph $\Gamma$ with color partitions Y and $Y'$ is said to be biregular if the valency of a vertex only depends on the color partition where it belongs to; see, for instance, [14].

For vertices $x_1,x_2,\ldots ,x_k\in X$ and nonnegative integers $i_1,i_2,\ldots ,i_k$ $(0\le i_1,i_2,\ldots ,i_k \le d)$, we define $\Gamma _{i_1,i_2,\ldots ,i_k}(x_1,x_2,\ldots ,x_k)=\bigcap _{\ell =1}^k \Gamma _{i_\ell }(x_\ell ).$ Assume that $y \in \Gamma _i(x)$ for some $0 \le i \le \varepsilon (x)$ and let z be a neighbor of y. Then, by the triangle inequality,

$$\begin{aligned} \partial (x, z) \in \left\{ i-1, i, i+1 \right\} , \end{aligned}$$

(2.1)

and so $z \in \Gamma _{i-1}(x) \cup \Gamma _i(x) \cup \Gamma _{i+1}(x)$. For $y \in \Gamma _i(x)$, we therefore define the following numbers:

$$\begin{aligned} a_i(x,y)&=\left| \Gamma _i(x)\cap \Gamma (y) \right| , \quad b_i(x,y)=\left| \Gamma _{i+1}(x)\cap \Gamma (y) \right| ,\\ c_i(x,y)&=\left| \Gamma _{i-1}(x)\cap \Gamma (y) \right| . \end{aligned}$$

We say that $x \in X$ is distance-regularized (or that $\Gamma$ is distance-regular around x) if the numbers $a_i(x,y), b_i(x,y)$ and $c_i(x,y)$ do not depend on the choice of $y \in \Gamma _i(x) \; (0 \le i \le \varepsilon (x))$. In this case, the numbers $a_i(x,y), b_i(x,y)$ and $c_i(x,y)$ are simply denoted by $a_i(x), b_i(x)$ and $c_i(x)$, respectively, and are called the intersection numbers of x. Observe that if x is distance-regularized and $\varepsilon (x)=d$, then $a_0(x)=c_0(x)=b_d(x)=0$, $b_0(x)= |\Gamma (x)|$ and $c_1(x)=1$. Note also that for every $1\le i \le d$ we have that $b_{i-1}(x)>0$ and $c_i(x)>0$, and that $a_i(x)=0$ if $\Gamma$ is bipartite. For convenience, we define $c_i(x)=b_i(x)=0$ for $i < 0$ and $i > d$.

A connected graph in which every vertex is distance-regularized is called a distance-regularized graph. A special case of such graphs are distance-regular graphs where all vertices have the same intersection array. Other examples are bipartite graphs in which vertices in the same color partition have the same intersection array, but which are not distance-regular. We call these graphs distance-biregular. It turns out that every distance-regularized graph is either distance-regular or distance-biregular (see [10]).

A connected bipartite graph $\Gamma$ with color partitions Y and $Y'$ is called distance semiregular with respect to Y if it is distance-regular around all vertices in Y, with the same parameters (i.e., there exist scalars $b_i$ and $c_i$ such that $b_i(x,y)=b_i$ and $c_i(x,y)=c_i$ for each $x\in Y$ and $y\in \Gamma _i(x)$). In this case, $\Gamma$ is biregular: each vertex in Y has valency $b_0$ and each vertex in $Y'$ has valency equal to $b_1+1$. Note that every distance-biregular graph is distance semiregular with respect to both color partitions Y and $Y'$.

The incidence graph of a design $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ is a $(\mathcal {P}, \mathcal {B})$-bipartite graph where the point $x \in \mathcal {P}$ is adjacent to the block $B\in \mathcal {B}$ if and only if x is incident with B. In this case, we observe that all points have eccentricity D, but the eccentricity of the blocks is not necessarily the same. If $D=1$, we observe that there exists a one-to-one correspondence between the incidence graph of 1-(1, 1, b) designs and bipartite distance-regularized graphs with vertices of eccentricity 1 (complete bipartite graphs $K_{1,b}$ with $b\ge 1$). If $D=2$, it is clear that there exists a one-to-one correspondence between the incidence graphs of 2-(v, v, b) designs and bipartite distance-regularized graphs with vertices of eccentricy 2 (complete bipartite graphs $K_{v,b}$ with $v\ge 2$, $b\ge 1$). The incidence graphs of symmetric 2-designs are precisely bipartite distance-regular graphs with vertices of eccentricity 3, which are well studied (see [1, 14]). The properties of incidence graphs of non-symmetric 2-designs were studied in [9]. In [9, Theorem 5.2], it is shown that there is a one-to-one correspondence between the incidence graphs of 2-designs and distance-semiregular graphs with distance-regularized vertices of eccentricity 3. Moreover, it turns out that quasi-symmetric 2-designs with one intersection number zero are exactly distance-biregular graphs with $D=3$ where every block has eccentricity $D'=4$. In this paper, we are interested in the incidence structures whose incidence graphs are bipartite distance-regularized graphs with all vertices of eccentricity 4.

3 Bipartite Distance-Regularized Graphs

In this section, we recall some results about bipartite distance-regularized graphs that we will find useful later in the rest of the paper. In particular, we introduce the 2-Y-homogeneous condition, which will be studied in more detail in Sect. 5.

Let us first observe some basic facts on bipartite distance-regularized graphs. Let $\Gamma$ denote a $(Y,Y')$-bipartite distance-regularized graph with vertex set X. By [10], we observe that $\Gamma$ is either a bipartite distance-regular graph ($\Gamma$ is regular and all of its vertices have the same intersection numbers) or $\Gamma$ is a distance-biregular graph ($\Gamma$ is not regular and vertices of the same bipartite class have the same intersection numbers). Pick now $x \in X$ and let $\varepsilon (x)$ denote the eccentricity of x. Since $\Gamma$ is bipartite, we have $a_i(x)=0$ for $0 \le i \le \varepsilon (x)$. Note that all vertices from Y ($Y'$, respectively) have the same eccentricity. We denote this common eccentricity by D ($D'$, respectively). We also observe that $|D-D'| \le 1$ and the diameter of $\Gamma$ equals $\max \{D,D'\}$. In addition, all vertices from Y ($Y'$, respectively) have the same valency k ($k'$, respectively). For $x \in Y$, $y \in Y'$ and an integer i we abbreviate $c_i:= c_i(x)$, $b_i:= b_i(x)$, $c'_i:= c_i(y)$ and $b'_i:= b_i(y)$. Observe that for $0 \le i \le D$, we have that $c_i+b_i=k$ if i is even, while $c_i+b_i=k'$ if i is odd. Moreover, if $k=k'$, it is not hard to see that $D=D'$ and the scalars $b_i=b_i'$, $c_i=c_i'$; see also [6, Lemma 1].

In the rest of this section, we recall some facts about the (almost) 2-Y-homogeneous condition for bipartite distance-regularized graphs.

Let $\Gamma$ denote a bipartite graph with vertex set X, color partitions Y, $Y'$, and assume that every vertex in Y has eccentricity $D\ge 3$. For $z\in X$ and a nonnegative integer i, recall that $\Gamma _{i}(z)$ denotes the set of vertices in X that are at distance i from z. Graph $\Gamma$ is almost 2-Y-homogeneous whenever for all $i \; (1\le i \le D-2)$ and for all $x\in Y$, $y \in \Gamma _2(x)$ and $z \in \Gamma _{i}(x)\cap \Gamma _i(y)$, the number of common neighbors of x and y which are at distance $i-1$ from z is independent of the choice of x, y and z. In addition, if the above condition holds also for the case $i=D-1$, then we say that $\Gamma$ is 2-Y-homogeneous. For $(Y,Y')$-bipartite distance-regularized graphs, we remark that the (almost) 2-Y-homogeneous condition generalizes the notion of (almost) 2-homogeneous distance-regular graphs which was well studied by Curtin and Nomura; see for more details [3,4,5, 12]. Moreover, the (almost) 2-Y-homogeneous condition in distance-biregular graphs was recently studied in [8] where the authors found necessary and sufficient conditions on the intersection array of $\Gamma$ for which the graph is (almost) 2-Y-homogeneous.

Assume that $\Gamma$ is a $(Y,Y')$-bipartite distance-regularized graph. Note that, if $D=2$ then $\Gamma$ is a complete bipartite graph with $D'\in \{1,2\}$ (if $D'=3$, then for any $x\in Y'$, $\Gamma _3(x)\subseteq Y$, which yields $D\ge 3$, a contradiction). Moreover, if $D=2$ then $\Gamma$ is 2-Y-homogeneous by definition. For the rest of the paper we assume that $D\ge 3$ (which also yields $D'\ge 3$—just use the same argument as in the previous sentence).

Suppose that $\Gamma$ has vertices of valency 2. If $\Gamma$ is regular then $\Gamma$ is a cycle of even length and so, $\Gamma$ is 2-Y-homogeneous and 2-$Y'$-homogeneous. Otherwise, by [11, Corollary 3.5], a graph $\Gamma$ with vertices of valency 2 is distance-biregular if and only if $\Gamma$ is either the complete bipartite graph $\Gamma =K_{2,r}$, or $\Gamma$ is the subdivision graph of a $(\kappa ,g)$-cage graph. In [8, Section 4], it was shown that a $(Y,Y')$-distance-biregular graph with $k'=2$ is 2-Y-homogeneous, and some combinatorial properties of such graphs were given. We then focus our attention on $(Y,Y')$-bipartite distance-regularized graphs with $k'\ge 3$.

In the next lemma, we introduce the scalars $p^i_{2,i}$ $(2 \le i \le D-1)$, which will play an important role in the study of the (almost) 2-Y-homogeneous condition of distance-regularized graphs.

Lemma 3.1

Let $\Gamma$ denote a $(Y,Y')$-bipartite distance-regularized graph. Let D denote the eccentricity of vertices from Y and assume $D\ge 3$. Pick a vertex $x\in Y$. For every integer $i \; (2 \le i \le D-1)$ and for $z \in \Gamma _i(x)$, the number of vertices which are at distance 2 from $x \in Y$ and at distance i from z depends only on the bipartite part Y but not on the choice of $x \in Y$ and $z \in \Gamma _i(x)$. Moreover,

$$\begin{aligned} |\Gamma _2(x)\cap \Gamma _i(z)|= \left\{ \begin{array}{ll} \frac{b_i(c_{i+1}-1)+c_i(b_{i-1}-1)}{c_2} &{}\quad \text{ if }\; i\;\text{ is } \text{ even, } \\ \frac{b'_i(c'_{i+1}-1)+c'_i(b'_{i-1}-1)}{c_2} &{}\quad \text{ if }\; i\;\text{ is } \text{ odd. } \end{array}\right. \end{aligned}$$

In this case, we simply write $p^i_{2,i}:=p^i_{2,i}(Y)=|\Gamma _2(x)\cap \Gamma _i(z)|$.

Proof

Recall that every bipartite distance-regularized graph is either a bipartite distance-regular graph or a distance-biregular graph. The proof now immediately follows from [1, Lemma 4.1.7] and [9, Lemma 3.1]. $\square$

We define certain scalars $\Delta _i$ $(2\le i\le \min \left\{ D-1, D'-1 \right\} )$, which can be computed from the intersection array of a given bipartite distance-regularized graph. These scalars play an important role, since from their values we can decide if a given bipartite distance-regularized graph is (almost) 2-Y-homogeneous or not.

Definition 3.2

Let $\Gamma$ denote a $(Y,Y')$-bipartite distance-regularized graph $\Gamma$. Let D ($D'$, respectively) denote the eccentricity of vertices from Y ($Y'$, respectively). Pick i $(1\le i \le \min \left\{ D-1, D'-1 \right\} )$, and with reference to Lemma 3.1, define the scalar $\Delta _i=\Delta _i(Y)$ in the following way:

$$\begin{aligned} \Delta _{i}= \left\{ \begin{array}{ll} (b_{i-1}-1)(c_{i+1}-1)-p^i_{2,i}(c_2'-1) &{}\quad \text{ if }\; i\; \text{ is } \text{ even, }\\ (b'_{i-1}-1)(c'_{i+1}-1)-p^i_{2,i}(c_2'-1) &{}\quad \text{ if }\;i\; \text{ is } \text{ odd. } \end{array}\right. \end{aligned}$$

We end this section pointing out the following results which we will find useful later to decide if a given bipartite distance-regularized graph has the (almost) 2-Y-homogeneous condition.

Theorem 3.3

With reference to Definition 3.2, let $\Gamma$ denote a $(Y,Y')$-bipartite distance-regularized graph with $k'\ge 3$ and $D\ge 3$. Then, the following are equivalent.

• (i) $\Delta _i=0$ $(2\le i\le \min \{D-1,D'-1\})$.

• (ii) For every i $(2\le i\le D-1)$, there exist $x \in Y$ and $y\in \Gamma _2(x)$ such that for all $z\in \Gamma _{i,i}(x,y)$ the number $|\Gamma _{1,1,i-1}(x,y,z)|$ is independent of the choice of z.

• (iii) $\Gamma$ is 2-Y-homogeneous.

Proof

Immediate from the definition of the 2-Y-homogeneous condition, [3, Theorem 13] and [8, Corollary 7.3]. $\square$

Theorem 3.4

With reference to Definition 3.2, let $\Gamma$ denote a $(Y,Y')$-bipartite distance-regularized graph with $k'\ge 3$ and $D\ge 3$. Then, the following are equivalent.

• (i) $\Delta _i=0$ $(2\le i\le D-2)$.

• (ii) For every i $(2\le i\le D-2)$, there exist $x \in Y$ and $y\in \Gamma _2(x)$ such that for all $z\in \Gamma _{i,i}(x,y)$ the number $|\Gamma _{1,1,i-1}(x,y,z)|$ is independent of the choice of z.

• (iii) $\Gamma$ is almost 2-Y-homogeneous.

Proof

Immediate from the definition of the 2-Y-homogeneous condition, [3, Theorem 13] and [8, Corollary 7.4]. $\square$

4 Special Partially Balanced Incomplete Block Designs

Quasi-symmetric special partially balanced incomplete block designs will be of particular interest for our work. In this section we introduce the concept of special partially balanced incomplete block designs and highlight some properties of such quasi-symmetric designs that are important for our research.

Let $\mathcal {D}$ be a 1-$(v,b,r,k,\lambda )$ design and let (s, t) be a pair of nonnegative integers. A flag (a non-flag) of $\mathcal {D}$ is a point-block pair (p, B) such that $p\in B$ ($p \notin B$). We say that $\mathcal {D}$ is a special partially balanced incomplete block design (SPBIBD for short) of type (s, t) if there are constants $\lambda _1$ and $\lambda _2$ with the following properties:

• (i) Any two points are contained in either $\lambda _1$ or $\lambda _2$ blocks.

• (ii) If a point-block pair (p, B) is a flag, then the number of points in B which occur with p in $\lambda _1$ blocks is s.

• (iii) If a point-block pair (p, B) is a non-flag, then the number of points in B which occur with p in $\lambda _1$ blocks is t.

In this case, we say that $\mathcal {D}$ is a $(v,b,r,k,\lambda _1, \lambda _2)$ SPBIBD of type (s, t).

Let $\mathcal {D}$ be a $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$. If $r=b$, then every point belongs to every block, which is a trivial case that we will not consider. Therefore, we will assume $r<b$. Also, if $t=0$, then every two different blocks do not interesect, and if $t=k$, then $\mathcal {D}$ is a 2-design. It is important to observe that in the instances mentioned above, the incidence graphs involved contain some vertices with eccentricity other than 4. Consequently, we will exclude consideration of such SPBIBDs in our analysis.

We are primarily interested in quasi-symmetric SPBIBDs, which we consider next.

The intersection numbers of a 1-$(v, k, \lambda )$ design are the cardinalities of the intersection of any two distinct blocks. Let x and y be nonnegative integers with $x< y$. A design $\mathcal {D}$ is called a (proper) quasi-symmetric design with intersection numbers x and y if any two distinct blocks of $\mathcal {D}$ intersect in either x or y points, and both intersection numbers are realized. That is, if $|B\cap B'|\in \{x,y\}$ for any pair of distinct blocks $B, B'$ and both intersection numbers occur. Furthermore, recall that the dual of a design $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ is the structure $\mathcal {D'}=(\mathcal {B}, \mathcal {P}, \mathcal {I'})$ such that $(B,p)\in \mathcal {I'}$ if and only if $(p,B)\in \mathcal {I}$ for every $(p,B)\in \mathcal {B} \times \mathcal {P}$. The dual of a quasi-symmetric SPBIBD is an SPBIBD (see [13, Theorem 4.39]).

Of particular interest for our work are quasi-symmetric SPBIBDs of type $(k-1,t)$ with intersection numbers $x=0$ and $y> 0$. The following applies to such SPBIBDs.

Lemma 4.1

Let $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ be a quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$. Then, $y\le t<k$. If $y>1$, then $t>y$ and $\lambda _1< t\lambda _1/y<r$.

Proof

Let (p, B) be a non-flag and let $B_1$ be a block such that $p \in B_1$ and $B_1$ intersects B in y points. From this it follows that the number of points in B which occur with p in $\lambda _1$ blocks is greater or equal than y, that is $y\le t$. Also $t<k$, since from $t=k$ it follows that every two points are together in $\lambda _1$ blocks.

Let $y>1$ and $y=t$. For a non-flag (p, B), let $\{p_1, p_2,\ldots ,p_t\} \subset B$ be the set of points in B which occur with p in $\lambda _1$ blocks. Every block $B_1$ containing p and $p_1$ will intersect B in points $p_1, p_2,\ldots ,p_t$ and, since $y=t$, it follows that $\lambda _1=1$. Also, in that case there are at least $2=\lambda _1+1$ blocks (B and $B_1$) containing $p_1$ and $p_2$, which is not possible. It follows that $t>y$. Furthermore, for a non-flag (p, B) there are exactly $t\lambda _1/y$ blocks containing p and intersecting B. Since $t>y,$ it follows $\lambda _1< t\lambda _1/y$. Also $t\lambda _1/y<r$, because on the contrary every two block will intersect. $\square$

Particular examples of quasi-symmetric SPBIBDs are partial geometries and generalized quadrangles. A partial geometry (r, k, t) is a 1-$(v,b,r, k, \lambda )$ design such that any two points are incident with at most one block, and for every non-flag (x, B), there exist exactly t blocks that are incident with x and intersect B $(1 \le t \le r, 1 \le t \le k)$. Notice that partial geometries are exactly quasi-symmetric (v, b, r, k, 1, 0) SPBIBDs of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$. A partial geometry for which $t=1$ is called a generalized quadrangle.

Finally, let us consider quasi-symmetric (v, b, r, k, 1, 0) SPBIBDs of type $(k-1,t)$ with intersection numbers $x=0$ and $y> 1$. For our further consideration, we need the following theorem.

Theorem 4.2

For the quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>1$, it follows that $k\ge 4$ and $r\ge 4$.

Proof

By Lemma 4.1, from $y>1$ we have $t>2$ and $k>3$. Furthermore, $0<\lambda _1<t\lambda _1/y<r$ implies that $r \ge 3$. If $r=3$ then $\lambda _1=1$, so $y=1$. We conclude that $r\ge 4.$ $\square$

5 The Incidence Graph of a SPBIBD

The incidence graph of a design $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ is a $(\mathcal {P}, \mathcal {B})$-bipartite graph where the point $x \in \mathcal {P}$ is adjacent to the block $B\in \mathcal {B}$ if and only if x is incident with B. In this section, we study some properties of the incidence graphs of certain SPBIBDs. The main result of this section is given in Theorem 5.6. We show that an incidence graph of a quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$ is a bipartite distance-regularized graph with all vertices having eccentricity 4. Additionally, we unveil a correlation between the intersection numbers of its vertices and the parameters of the corresponding SPBIBD. This pivotal outcome will play a crucial role in Sect. 6, where we embark on characterizing bipartite distance-regularized graphs with all vertices of eccentricity 4. The 2-$\mathcal {P}$-homogeneous and 2-$\mathcal {B}$-homogeneous conditions in these graphs are also studied. Obtained results will be used in Sect. 6 for classification of 2-Y-homogeneous bipartite distance-regularized graphs with $D=4$ and $c'_2=1$.

Before we restrict our attention to quasi-symmetric SPBIBDs, we will prove some properties of all $(v,b,r,k,\lambda _1, 0)$ SPBIBDs of type $(k-1,t)$ that we will need later.

Lemma 5.1

Let $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ be a $(v,b,r,k,\lambda _1, 0)$ SPBIBD of type $(k-1,t)$ where $0<t<k$. Let $\Gamma$ denote the incidence graph of $\mathcal {D}$. Then, every vertex $p \in \mathcal {P}$ has eccentricity 4 in $\Gamma$.

Proof

Notice that for any $p\in \mathcal {P}$, there are r blocks containing p and so, which are at distance 1 from p. Moreover, since $r<b$ there exists a block $B \in \mathcal {B}$ such that (p, B) is not a flag.

Let S denote the set of all points incident with B which are at distance greater than 2 from p. We observe that S is nonempty and has size $k-t<k$. Pick $q\in S$. Since $\Gamma$ is bipartite, in this case we have that $\partial (p,q)\ge 4$. Note also that there are t points in B which are at distance 2 from p. Let w be such a point. For a block $B'$ containing both p and w, it follows that $\left[ p, B', w, B, q\right]$ is a pq-path of length 4, meaning that $\partial (p,q)=4$ and $\partial (p,B)=3$. Hence, from the above comments, every point $p\in \mathcal {P}$ has eccentricity 4. $\square$

Lemma 5.2

Let $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ be a $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$, where $0<t<k.$ Let $\Gamma$ denote the incidence graph of $\mathcal {D}$. Then, every vertex $p \in \mathcal {P}$ is distance-regularized. Moreover, $\Gamma$ is distance semiregular with respect to $\mathcal {P}$ with the following intersection numbers:

$$\begin{aligned} c_0= & {} 0, \quad c_{1}=1, \quad c_{2}=\lambda _1, \quad c_{3}=t, \quad c_{4}=r. \\ b_0= & {} r, \quad b_{1}=k-1, \quad b_{2}=r-\lambda _1, \quad b_{3}=k-t \quad b_{4}=0. \end{aligned}$$

Proof

Note that $\Gamma$ is bipartite with bipartitions $\mathcal {P}$ and $\mathcal {B}$. As every point is contained in r blocks and every block has size k, it is easy to see that $\Gamma$ is (r, k)-biregular. We notice that every pair of points either are contained in exactly $\lambda _1$ blocks or are not contained in any block. Moreover, by Lemma 5.1 every point in $\mathcal {P}$ has eccentricity equal to 4.

We will now show that every point in $\mathcal {P}$ is distance-regularized. Pick $p\in \mathcal {P}$. For every $z\in \Gamma _i(p) \; (0 \le i \le 4)$, consider the numbers $b_i(p,z)$, $a_i(p,z)$ and $c_i(p,z)$. Recall that $a_i(p,z)=0$ as the graph $\Gamma$ is bipartite. Note that $b_0(p,z)=|\Gamma (p)|$, $c_0(p,z)=0$, $c_1(p,z)=1$ and $b_4(p,z)=0$. As every block has size k, every block in $\Gamma (p)$ contains $k-1$ points different from p and so, $b_1(p,z)=k-1$. Similarly, as every pair of points that appears in a block is contained in $\lambda _1$ blocks we have $c_2(p,z)=\lambda _1$ and, since every point is contained in r blocks, we have $b_2(p,z)=r-\lambda _1$. Moreover, for a block $z\in \Gamma _3(p)$ we have t points in $z\in \mathcal {B}$ which occur with p in $\lambda _1$ blocks. This yields $c_3(p,z)=t$ and $b_3(p,z)=k-t$, since every block has size k. Furthermore, $c_4(p,z)=r$ as every point appears in r blocks. Thus, the numbers $b_i(p,z)$, $a_i(p,z)$ and $c_i(p,z)$ do not depend on the choice of $z\in \Gamma _i(p) \; (0 \le i \le 4)$, and $\Gamma$ is distance-regular around p. It follows from the above comments that $\Gamma$ is distance semiregular with respect to $\mathcal {P}$. $\square$

Now we restrict our attention to quasi-symmetric SPBIBDs.

Lemma 5.3

Let $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ be a quasi-symmetric $(v,b,r,k, \lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$. Then, for every $B \in \mathcal {B}$ there exist $B_1, B_2 \in \mathcal {B}$ such that $|B\cap B_1|=0$ and $|B\cap B_2|=y$.

Proof

Let $B \in \mathcal {B}$. Since $k<v$, there exists $p \in \mathcal {P}$ such that (p, B) is a non-flag. Note also that there are t points in B which are at distance 2 from p. Let q be such a point. Then, there exists a block $B_2 \in \mathcal {B}$ such that $\{p,q\}\subseteq B_2$. Thus, $q \in B\cap B_2$ and therefore, $|B\cap B_2|=y$.

Let $\mathcal {D}'$ denote the dual of $\mathcal {D}$. Since $\mathcal {D}$ is quasi-symmetric, $\mathcal {D}'$ is an SPBIBD, let say of type $(s',t')$. Now, suppose that B intersects all the other blocks in $\mathcal {B}$. This means that in the incidence graph of $\mathcal {D}'$ there exists a point $p'$ which is at distance 2 from any other point. If p is contained in all blocks of $\mathcal {D}'$, then $k=v$, which is not possible. So, there exists a block $B'$ of $\mathcal {D}'$ such that the pair $(p,B')$ is a non-flag. Since $p'$ is at distance 2 from any other point then the number of points in $B'$ that are at distance 2 from $p'$ equals the size of a block. That is, $t'=r$.

Now, let $q'\ne p'$ be a point of $\mathcal {D}'$. Then, any other point $q''$ in $\mathcal {D}'$ is either in a block with $q''$ or not. In the first case, the distance between $q'$ and $q''$ is 2. In the second case, there exists a block $B''$ of $\mathcal {D}'$ such that $q'' \in B''$ and the pair $(q', B'')$ is a non-flag. Since $t'$ equals the size of the block in $\mathcal {D}'$, we have that $q''$ is at distance 2 from $q'$. This shows that $q'$ is at distance two from any other point. Therefore, any two points of $\mathcal {D}'$ are at distance 2. In other words, any two blocks of $\mathcal {D}$ intersect, which contradicts $x=0$. Hence, there must be a block $B_1\in \mathcal {B}$ such that $B\cap B_1=\emptyset$. This finishes the proof. $\square$

Corollary 5.4

Let $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ be a quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$. Then, $t<r$.

Proof

Assume that $t=r$. Pick $B\in \mathcal {B}.$ Then for every $p \in \mathcal {P}$ and $B' \in \mathcal {B}$ such that (p, B) is a non-flag and $(p,B')$ is a flag, it follows that $\left| B\cap B'\right| =y$. Hence, there is no $B_1 \in \mathcal {B}$ such that $\left| B\cap B_1\right| =0$, contradicting Lemma 5.3. $\square$

Using the two preceding results, we proceed to establish that $D' = 4$ holds true for the incidence graph corresponding to a quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$.

Lemma 5.5

Let $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ be a quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$. Let $\Gamma$ denote the incidence graph of $\mathcal {D}$. Then, every vertex $B \in \mathcal {B}$ has eccentricity 4 in $\Gamma$.

Proof

Recall that $\Gamma$ is the $(\mathcal {P}, \mathcal {B})$-bipartite graph where the point $p \in \mathcal {P}$ is adjacent to the block $B\in \mathcal {B}$ if and only if p is incident with B. Moreover, as $\mathcal {D}$ is quasi-symmetric, any two distinct blocks of $\mathcal {D}$ intersect in either $x=0$ or $y>0$ points, and both intersection numbers are realized. By Lemma 5.3, we also know that for every $B \in \mathcal {B}$ there is a block that intersects B and another block that does not intersect B.

Pick $B \in \mathcal {B}$ and let $B' \in \mathcal {B}$. If $B\cap B'\ne \emptyset$ then B and $B'$ have a point in common, showing that $\partial (B,B')=2$. Assume next that $B\cap B'=\emptyset$. Pick $p\in B$. Since $(p,B')$ is not a flag there exist t points in $B'$ which are at distance 2 from p. Let w be such a point. Notice $w\notin B$ since B and $B'$ are disjoint. So, for a block $B''$ containing both p and w, it follows that $\left[ B, p, B'', w, B'\right]$ is a path of length 4, meaning that $\partial (B,B')=4$.

Pick now $p \in \mathcal {P}$ and $B\in \mathcal {B}$. If $p\in B$ we have that $\partial (p,B)=1$. Suppose next that $p \notin B$. Since $\Gamma$ is bipartite we observe that $\partial (p,B)$ is odd and so, $\partial (p,B)\ge 3$. Moreover, by Lemma 5.1, every point in $\mathcal {P}$ has eccentricity 4 and so $\partial (p,B)\le 4$. This shows that $\partial (p,B)= 3$ if p does not belong to B. Thus, the eccentricity of any block in $\Gamma$ is equal to 4. $\square$

The following theorem is the main result of this section. It characterizes incidence graphs of quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBDs of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$ and connects them with bipartite distance-regularized graphs with $D=D'=4$.

Theorem 5.6

Let $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ be a quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$. Let $\Gamma$ denote the incidence graph of $\mathcal {D}$. Then, $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph. Moreover, every vertex $p \in \mathcal {P}$ has eccentricity equals 4 and the following intersection numbers:

$$\begin{aligned} c_0= & {} 0, \quad c_{1}=1, \quad c_{2}=\lambda _1, \quad c_{3}=t, \quad c_{4}=r. \\ b_0= & {} r, \quad b_{1}=k-1, \quad b_{2}=r-\lambda _1, \quad b_{3}=k-t \quad b_{4}=0. \end{aligned}$$

In addition, every vertex $B \in \mathcal {B}$ has eccentricity equals 4 and the following intersection numbers:

$$\begin{aligned} c_0'= & {} 0,\quad c'_{1}=1, \quad c'_{2}=y, \quad c'_{3}=\frac{t\lambda _1}{y}, \quad c'_{4}=k. \\ b_0'= & {} k, \quad b'_{1}=r-1, \quad b_{2}'=k-y, \quad b'_{3}=r-\frac{t\lambda _1}{y}, \quad b'_{4}=0. \end{aligned}$$

Proof

Since $\mathcal {D}$ is a $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ where $0< t<k$, by Lemma 5.2, every vertex in $\mathcal {P}$ has eccentricity 4 and $\Gamma$ is distance semiregular with respect to $\mathcal {P}$, where the intersection numbers for every $p\in \mathcal {P}$ can be computed as in the proof of Lemma 5.2. Moreover, by Lemma 5.5 the eccentricity of every block in $\Gamma$ equals 4.

We will now show that every block in $\Gamma$ is distance-regularized. Pick $B\in \mathcal {B}$. For every $z\in \Gamma _i(B) \; (0 \le i \le 4)$, consider the numbers $b'_i(B,z)$, $a'_i(B,z)$ and $c'_i(B,z)$. Recall that $a'_i(B,z)=0$, as the graph $\Gamma$ is bipartite. Note that $b'_0(B,z)=|\Gamma (B)|$, $c'_0(B,z)=0$, $c'_1(B,z)=1$ and $b'_4(B,z)=0$. As every two blocks which are at distance 2 have y common neighbors, we have $c'_2(B,z)=y$. Suppose for the moment that $z\in \Gamma _3(B)$. We will count the number of paths of length 3 between B and z in two different ways. Firstly, observe that z has $c'_3(B,z)$ neighbors in $\Gamma _2(B)$ and each of these neighbors is adjacent to exactly y vertices in $\Gamma (B)$, since every two intersecting blocks have y points in common. Secondly, note that there are t points in B which occur with p in $\lambda _1$ blocks. Thus, we have $c'_3(B,z)y=t\lambda _1$. Also, as each block contains k points, we have $c'_4(B,z)=k$. Furthermore, since every point occurs in r blocks and each block has size k, it follows from the above comments the numbers $b'_i(B,z)$ and $c'_i(B,z)$ do not depend on the choice of $z\in \Gamma _i(B) \; (0 \le i \le 4)$. Hence, $\Gamma$ is distance-regular around B. If $k=r$ then by [6, Lemma 1] we have that $\lambda _1=y$ and so, $\Gamma$ is a bipartite distance-regular graph with diameter 4. Otherwise, $\Gamma$ is a distance-biregular graph with $D=D'=4$. This finishes the proof. $\square$

In the remainder of this section, we study the 2-Y-homogeneous condition of quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBDs of type $(k-1,t)$ with one intersection number equal to 0.

5.1 The (Almost) 2-$\mathcal {P}$-Homogeneous Condition

According to Theorem 5.6, the incidence graph of certain SPBIBDs is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph with all vertices of eccentricity equal to four. Here, we explore the (almost) 2-$\mathcal {P}$-homogeneous condition of the incidence graph of such SPBIBD.

In the next two propositions, we consider the case of quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBDs of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$. These results will be used in the proof of Theorem 6.5.

Proposition 5.7

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$. The following (i),(ii) are equivalent:

• (i) Each block has size 2.

• (ii) $\Gamma$ is isomorphic to the subdivision graph of a complete bipartite graph $K_{r,r}=(X,\mathcal {R})$ with $r \ge 2$ and $X=\mathcal {P}$.

In this case, $\Gamma$ is 2-$\mathcal {P}$-homogeneous.

Proof

Suppose that each block has size 2. We observe that $k=2$ and $y=\lambda _1=1$. By Lemma 4.1, we also have that $1=y\le t <k=2$ and so, $t=1$. By Theorem 5.6, we thus have that $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph where every point and every block has eccentricity equal to four. Since every vertex has eccentricity 4 we have that $r>1$. If $r=2$ then $\Gamma$ is a distance-regular graph of diameter 4 and valency 2; i.e., $\Gamma$ is the subdivision graph of the complete bipartite graph $K_{2,2}$. Otherwise, $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-distance-biregular graph. Moreover, the intersection numbers of every point are $b_0=r$, $c_i=1 \; (1\le i \le 3)$ and $c_4=r$. It follows from [8, Theorem 4.2] that $\Gamma$ is isomorphic to the subdivision graph of an (r, 4)-cage, i.e., a complete bipartite graph $K_{r,r}=(X,\mathcal {R})$ with $r \ge 2$ and $X=\mathcal {P}$. Conversely, we notice that $\Gamma$ is a $(\mathcal {P},\mathcal {B})$-bipartite graph with $|\mathcal {P}|=2r$ and $|\mathcal {B}|=r^2$ where every point in $\mathcal {P}$ has valency r while every block in $\mathcal {B}$ has valency 2. To prove our last claim, observe that from [8, Theorem 4.2], for all $p\in \mathcal {P}$ and $q \in \Gamma _2(p)$, the sets $\Gamma _{2,2}(p, q)$ are empty. Thus, for all $i \; (2 \le i \le 3)$ and for all $p\in \mathcal {P}$, $q \in \Gamma _2(p)$, and $z \in \Gamma _{2,2}(p, q)$, the number $|\Gamma _{i-1}(z) \cap \Gamma _{1,1}(x, y)|$ equals 0. This shows that $\Gamma$ is 2-$\mathcal {P}$-homogeneous. $\square$

Proposition 5.8

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$. Assume that $k\ge 3$. Then, the following hold:

• (i) $\Gamma$ is almost 2-$\mathcal {P}$-homogeneous if and only if $\mathcal {D}$ is a generalized quadrangle.

• (ii) $\Gamma$ is not 2-$\mathcal {P}$-homogeneous.

Proof

By Theorem 5.6, we observe that $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph where every point and every block has eccentricity equal to four. Moreover, the intersection arrays of $\Gamma$ can be computed in terms of the parameters of $\mathcal {D}$. Therefore, to analyze the (almost) 2-$\mathcal {P}$-homogeneous condition of $\Gamma$, it suffices to compute the scalars $\Delta _2(\mathcal {P})$ and $\Delta _3(\mathcal {P})$, as defined in Definition 3.2. For $(0\le i \le 4)$, let $c_i, b_i$ and $c'_i, b'_i$ denote the intersection numbers of the points and the blocks, respectively, as shown in Theorem 5.6. Since $c_2'=y=1$, it turns out that

$$\begin{aligned} \Delta _2(\mathcal {P})&=(b_1-1)(c_3-1)=(k-2)(t-1) , \end{aligned}$$

(5.1)

$$\begin{aligned} \Delta _3(\mathcal {P})&=(b_2'-1)(c_4'-1)=(k-2)(k-1) . \end{aligned}$$

(5.2)

By Theorem 3.4, we have that $\Gamma$ is almost 2-$\mathcal {P}$-homogeneous if and only if $\Delta _2(\mathcal {P})=0$. If $\mathcal {D}$ is a generalized quadrangle then $t=1$ which shows that $\Delta _2(\mathcal {P})=0$. Conversely, if $\Delta _2(\mathcal {P})=0$ then $t=1$, since $k>2$. This means that $\mathcal {D}$ is a generalized quadrangle if and only if $\Delta _2(\mathcal {P})=0$. So, (i) follows. Moreover, from (5.2) the scalar $\Delta _3(\mathcal {P})>0$ and so, by Theorem 3.3 we have that $\Gamma$ is not 2-$\mathcal {P}$-homogeneous. $\square$

The study of (almost) 2-$\mathcal {P}$-homogeneous condition of quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBDs of type $(k-1,t)$ with intersection numbers $x=0$ and $y>1$ is much more complicated. Some properties of such designs are given in Proposition 5.9 and Proposition 5.10.

Proposition 5.9

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>1$. In this case, $\Gamma$ is almost 2-$\mathcal {P}$-homogeneous if and only if

$$\begin{aligned} k=\frac{(y-1)(r-\lambda _1)(t-1)}{\lambda _1(t-y)}+2. \end{aligned}$$

Proof

By Theorem 5.6, we observe that $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph where every point and every block has eccentricity equal to four. Moreover, the intersection arrays of $\Gamma$ can be computed in terms of the parameters of $\mathcal {D}$. Therefore, to analyze the (almost) 2-$\mathcal {P}$-homogeneous condition of $\Gamma$, it suffices to compute the scalars $p^2_{2,2}(\mathcal {P})$, as shown in Lemma 3.1, and $\Delta _2(\mathcal {P})$, as defined in Definition 3.2. For $(0\le i \le 4)$, let $c_i, b_i$ and $c'_i, b'_i$ denote the intersection numbers of the points and the blocks, respectively, as shown in Theorem 5.6. Therefore, by Lemma 3.1 it follows that

$$\begin{aligned} p^2_{2,2}(\mathcal {P})=\frac{(r-\lambda _1)(t-1)+\lambda _1(k-2)}{\lambda _1}. \end{aligned}$$

(5.3)

Then, by Definition 3.2 and (5.3) we have that

$$\begin{aligned} \Delta _2(\mathcal {P})=(k-2)(t-1)-(y-1)\frac{(r-\lambda _1)(t-1)+\lambda _1(k-2)}{\lambda _1}. \end{aligned}$$

(5.4)

By Theorem 3.4, we have that $\Gamma$ is almost 2-$\mathcal {P}$-homogeneous if and only if $\Delta _2(\mathcal {P})=0$. From (5.4), it is easy to see that $\Delta _2(\mathcal {P})=0$ if and only if

$$\begin{aligned} \lambda _1(k-2)(t-y)=(y-1)(r-\lambda _1)(t-1). \end{aligned}$$

(5.5)

Since $y>1$, by Lemma 4.1 and Theorem 4.2 we also observe that $t>y$ and $k\ge 4$. Therefore, from (5.5) it is easy to see that

$$\begin{aligned} k-2=\frac{(y-1)(r-\lambda _1)(t-1)}{\lambda _1(t-y)}. \end{aligned}$$

The claim now immediately follows from Theorem 3.4. $\square$

Proposition 5.10

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>1$. In this case, $\Gamma$ is 2-$\mathcal {P}$-homogeneous if and only if

$$\begin{aligned} k=\frac{(y-1)(r-\lambda _1)(t-1)}{\lambda _1(t-y)}+2, \quad t=\frac{(k-1)\left[ r(y-1)-\lambda _1(k-y-1) \right] }{\lambda _1(y-1)}. \end{aligned}$$

Proof

By Theorem 5.6, we observe that $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph where every point and every block has eccentricity equal to four. Moreover, the intersection arrays of $\Gamma$ can be computed in terms of the parameters of $\mathcal {D}$. Therefore, to analyze the 2-$\mathcal {P}$-homogeneous condition of $\Gamma$, it suffices to compute the scalars $p^3_{2,3}(\mathcal {P})$, as shown in Lemma 3.1, and $\Delta _3(\mathcal {P})$, as defined in Definition 3.2. For $(0\le i \le 4)$, let $c_i, b_i$ and $c'_i, b'_i$ denote the intersection numbers of the points and the blocks, respectively, as shown in Theorem 5.6. Therefore, by Lemma 3.1 it follows that

$$\begin{aligned} p^3_{2,3}(\mathcal {P})= \frac{\left( r-\frac{t\lambda _1}{y}\right) (k-1)+\frac{t\lambda _1}{y}(k-y-1) }{\lambda _1} . \end{aligned}$$

(5.6)

Then, by Definition 3.2 and (5.6) we have that

$$\begin{aligned} \Delta _3(\mathcal {P})=(k-y-1)(k-1)-(y-1)\frac{\left( r-\frac{t\lambda _1}{y}\right) (k-1)+\frac{t\lambda _1}{y}(k-y-1) }{\lambda _1}. \end{aligned}$$

(5.7)

By Theorems 3.3 and 3.4, we have that $\Gamma$ is 2-$\mathcal {P}$-homogeneous if and only if $\Gamma$ is almost 2-$\mathcal {P}$-homogeneous and $\Delta _3(\mathcal {P})=0$. From (5.7), it is easy to see that $\Delta _3(\mathcal {P})=0$ if and only if

$$\begin{aligned} r(k-1)-t\lambda _1=\frac{\lambda _1(k-y-1)(k-1)}{y-1}. \end{aligned}$$

(5.8)

Therefore, from (5.8) it is easy to see that

$$\begin{aligned} t=\frac{(k-1)\left[ r(y-1)-\lambda _1(k-y-1) \right] }{\lambda _1(y-1)}. \end{aligned}$$

The claim now immediately follows from the above comments and Proposition 5.9. $\square$

As we delve into our research, Propositions 5.9 and 5.10 have surfaced, revealing a compelling problem that demands further investigation.

Problem 5.11

Determine the existence of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y > 1$ whose incidence graph is (almost) 2-$\mathcal {P}$-homogeneous.

5.2 The (Almost) 2-$\mathcal {B}$-Homogeneous Condition

This section investigates the (almost) 2-$\mathcal {B}$-homogeneous properties of the incidence graph of certain SPBIBDs. Based on Theorem 5.6, this incidence graph is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph, with all vertices having eccentricity equal to four. Similarly to the previous section, we classify 2-$\mathcal {B}$-homogeneous incidence graphs of quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBDs of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$ completely, and then consider the (almost) 2-$\mathcal {B}$-homogeneous condition for the case $y> 1$.

Proposition 5.12

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$. The following (i), (ii) are equivalent:

• (i) Each point appears in exactly 2 blocks.

• (ii) $\Gamma$ is isomorphic to the subdivision graph of a complete bipartite graph $K_{k,k}$ and $b=2k$.

In this case, $\Gamma$ is 2-$\mathcal {B}$-homogeneous.

Proof

Similar to the proof of Proposition 5.7. $\square$

Proposition 5.13

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$. Assume that $r\ge 3$. Then the following hold:

• (i) $\Gamma$ is almost 2-$\mathcal {B}$-homogeneous if and only if $\mathcal {D}$ is a generalized quadrangle.

• (ii) $\Gamma$ is not 2-$\mathcal {B}$-homogeneous.

Proof

By Theorem 5.6, we observe that $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph where every point and every block has eccentricity equal to four. Moreover, the intersection arrays of $\Gamma$ can be computed in terms of the parameters of $\mathcal {D}$. Therefore, to analyze the (almost) 2-$\mathcal {B}$-homogeneous of $\Gamma$, it suffices to compute the scalars $\Delta _2(\mathcal {B})$ and $\Delta _3(\mathcal {B})$, as defined in Definition 3.2. For $(0\le i \le 4)$, let $c_i, b_i$ and $c'_i, b'_i$ denote the intersection numbers of the points and the blocks, respectively, as shown in Theorem 5.6. Since $c_2'=y=1$, it follows that $c_2=\lambda _1=1$. We thus have that

$$\begin{aligned} \Delta _2(\mathcal {B})&=(b'_1-1)(c'_3-1)=(r-2)(t-1) , \end{aligned}$$

(5.9)

$$\begin{aligned} \Delta _3(\mathcal {B})&=(b_2-1)(c_4-1)=(r-2)(r-1) . \end{aligned}$$

(5.10)

By Theorem 3.4, we have that $\Gamma$ is almost 2-$\mathcal {B}$-homogeneous if and only if $\Delta _2(\mathcal {B})=0$. If $\mathcal {D}$ is a generalized quadrangle then $t=1$ which shows that $\Delta _2(\mathcal {B})=0$. Conversely, if $\Delta _2(\mathcal {B})=0$ then $t=1$, since $r>2$. This means that $\mathcal {D}$ is a generalized quadrangle if and only if $\Delta _2(\mathcal {B})=0$. So, (i) follows. To prove the second part of our claim, we observe from (5.10) that the scalar $\Delta _3(\mathcal {B})>0$ and so, by Theorem 3.3 we have that $\Gamma$ is not 2-$\mathcal {B}$-homogeneous. This finishes the proof. $\square$

Proposition 5.14

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>1$. In this case, $\Gamma$ is almost 2-$\mathcal {B}$-homogeneous if and only if

$$\begin{aligned} r=\frac{(k-y)\left( \frac{t\lambda _1}{y}-1\right) (\lambda _1-1)}{\lambda _1(t-y)}+2. \end{aligned}$$

Proof

By Theorem 5.6, we observe that $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph where every point and every block has eccentricity equal to four. Moreover, the intersection arrays of $\Gamma$ can be computed in terms of the parameters of $\mathcal {D}$. Therefore, to analyze the (almost) 2-$\mathcal {B}$-homogeneous condition of $\Gamma$, it suffices to compute the scalars $p^2_{2,2}(\mathcal {B})$, as shown in Lemma 3.1, and $\Delta _2(\mathcal {B})$, as defined in Definition 3.2. For $(0\le i \le 4)$, let $c_i, b_i$ and $c'_i, b'_i$ denote the intersection numbers of the points and the blocks, respectively, as shown in Theorem 5.6. Therefore, by Lemma 3.1 it follows that

$$\begin{aligned} p^2_{2,2}(\mathcal {B})=\frac{(k-y)\left( \frac{t\lambda _1}{y}-1\right) +y(r-2)}{y}. \end{aligned}$$

(5.11)

Then, by Definition 3.2 and (5.3) we have that

$$\begin{aligned} \Delta _2(\mathcal {B})=(r-2)\left( \frac{t\lambda _1}{y}-1 \right) -(\lambda _1-1)\frac{(k-y)\left( \frac{t\lambda _1}{y}-1\right) +y(r-2)}{y}. \end{aligned}$$

(5.12)

By Theorem 3.4, we have that $\Gamma$ is almost 2-$\mathcal {B}$-homogeneous if and only if $\Delta _2(\mathcal {B})=0$. From (5.12), it is easy to see that $\Delta _2(\mathcal {B})=0$ if and only if

$$\begin{aligned} \lambda _1(r-2)(t-y)=(k-y)\left( \frac{t\lambda _1}{y}-1\right) (\lambda _1-1). \end{aligned}$$

(5.13)

Since $y>1$, by Lemma 4.1 we also observe that $t>y$. Therefore, from (5.13) it is easy to see that

$$\begin{aligned} r-2=\frac{(k-y)\left( \frac{t\lambda _1}{y}-1\right) (\lambda _1-1)}{\lambda _1(t-y)}. \end{aligned}$$

The claim now immediately follows from Theorem 3.4. $\square$

Proposition 5.15

Let $\Gamma$ denote the incidence graph of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y>1$. In this case, $\Gamma$ is 2-$\mathcal {B}$-homogeneous if and only if

$$\begin{aligned} r=\frac{(k-y)\left( \frac{t\lambda _1}{y}-1\right) (\lambda _1-1)}{\lambda _1(t-y)}+2, \quad t=\frac{(r-1)\left[ k(\lambda _1-1)-y(r-\lambda _1-1) \right] }{\lambda _1(\lambda _1-1)}. \end{aligned}$$

Proof

By Theorem 5.6, we observe that $\Gamma$ is a $(\mathcal {P}, \mathcal {B})$-bipartite distance-regularized graph where every point and every block has eccentricity equal to four. Moreover, the intersection arrays of $\Gamma$ can be computed in terms of the parameters of $\mathcal {D}$. Therefore, to analyze the 2-$\mathcal {B}$-homogeneous condition of $\Gamma$, it suffices to compute the scalars $p^3_{2,3}(\mathcal {B})$, as shown in Lemma 3.1, and $\Delta _3(\mathcal {B})$, as defined in Definition 3.2. For $(0\le i \le 4)$, let $c_i, b_i$ and $c'_i, b'_i$ denote the intersection numbers of the points and the blocks, respectively, as shown in Theorem 5.6. Therefore, by Lemma 3.1 it follows that

$$\begin{aligned} p^3_{2,3}(\mathcal {B})= \frac{(k-t)(r-1)+t(r-\lambda _1-1) }{y} . \end{aligned}$$

(5.14)

Then, by Definition 3.2 and (5.14) we have that

$$\begin{aligned} \Delta _3(\mathcal {B})=(r-\lambda _1-1)(r-1)-(\lambda _1-1)\frac{(k-t)(r-1)+t(r-\lambda _1-1) }{y}. \end{aligned}$$

(5.15)

By Theorems 3.3 and 3.4, we have that $\Gamma$ is 2-$\mathcal {B}$-homogeneous if and only if $\Gamma$ is almost 2-$\mathcal {B}$-homogeneous and $\Delta _3(\mathcal {B})=0$. From (5.15), it is easy to see that $\Delta _3(\mathcal {B})=0$ if and only if

$$\begin{aligned} \left[ (k-t)(r-1)+t(r-\lambda _1-1)\right] (\lambda _1-1)=y(r-1)(r-\lambda _1-1). \end{aligned}$$

(5.16)

Therefore, from (5.16) it is easy to see that

$$\begin{aligned} t=\frac{(r-1)\left[ k(\lambda _1-1)-y(r-\lambda _1-1) \right] }{\lambda _1(\lambda _1-1)}. \end{aligned}$$

The claim now immediately follows from the above comments and Proposition 5.14. $\square$

In the course of our research, Propositions 5.14 and 5.15 have brought forth a natural problem that warrants further investigation.

Problem 5.16

Determine the existence of a $\mathcal {D}=(\mathcal {P}, \mathcal {B}, \mathcal {I})$ quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y > 1$ whose incidence graph is (almost) 2-$\mathcal {B}$-homogeneous.

6 Distance-Semiregular Graphs and SPBIBDs

In this section, we will prove the main result of our work. In other words, we will give a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k,\lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y> 0$ and bipartite distance-regularized graphs with vertices of eccentricity 4.

Initially, we prove that every distance-semiregular graph is the incidence graph of a SPBIBD, and relate its intersection numbers to the parameters of the corresponding SPBIBD. Subsequently, we employ this result to show a one-to-one correspondence between the incidence graphs of certain SPBIBDs and distance semiregular graphs with $D=4$.

Lemma 6.1

Let $\Gamma$ be a $(Y,Y')$-distance semiregular graph with respect to Y. Assume every vertex in Y has eccentricity $D=4$. Let $b_i, c_i \; (0\le i \le 4)$ denote the intersection numbers of every vertex in Y. Then, $\Gamma$ is the incidence graph of a $(1+\frac{b_0b_1}{c_2}+\frac{b_0b_1b_2b_3}{c_2c_3c_4},b_0+\frac{b_0b_1b_2}{c_2c_3},b_0,b_0', c_2,0)$ SPBIBD of type $(b_1,c_3)$.

Proof

Since $\Gamma$ is a $(Y,Y')$-distance semiregular graph with respect to Y, it is $(b_0,b_0')$-biregular where $b_0'$ denotes the valency of a vertex in $Y'$. We next consider the combinatorial incidence structure $\mathcal {D}=(Y, Y', \mathcal {I})$, with point set Y, block set $Y'$ and incidence $\mathcal {I}$. We will prove that $\mathcal {D}$ is a SPBIBD. As $\Gamma$ is $(b_0,b_0')$-biregular, it follows that every block has size $b_0'$ and every point is contained in $b_0$ blocks. Moreover, since $\Gamma$ is bipartite and every point has eccentricity 4, for every $p\in Y$ we have that $|Y|=|\Gamma _0(p)|+|\Gamma _2(p)|+|\Gamma _4(p)|$ and $|Y'|=|\Gamma _1(p)|+|\Gamma _3(p)|$. The distance-regularity property around p gives us that

$$\begin{aligned} |Y|=1+\frac{b_0b_1}{c_2}+\frac{b_0b_1b_2b_3}{c_2c_3c_4}, \quad |Y'|=b_0+\frac{b_0b_1b_2}{c_2c_3}. \end{aligned}$$

We next claim that every two points are contained either in $\lambda _1=c_2$ blocks or in $\lambda _2=0$ blocks. To prove our claim, pick $p\in Y$. Note that p has eccentricity 4 and since $\Gamma$ is bipartite, any other point different from p is at distance 2 or at distance 4 from p. Let $q\in \Gamma _2(p)$. Then, we need to count the number of common neighbors of p and q. We observe that $|\Gamma (p)\cap \Gamma (q)|=c_2$. Now, for $q\in \Gamma _4(p)$ we observe that there is no block containing both p and q. This proves our claim.

Next, pick $p\in Y$ and $B\in Y'$. If p is a neighbor of B we observe that $|\Gamma (B)\cap \Gamma _2(p)|=b_1$. So, if (p, B) is a flag, then the number of points in B which occur with p in $c_2$ blocks is $b_1$. Similarly, if (p, B) is a non-flag, then $B\in \Gamma _3(p)$ and the number of points in B which occur with p in $c_2$ blocks is $|\Gamma (B)\cap \Gamma _2(p)|=c_3$. Therefore, $\mathcal {D}$ is a SPBIBD of type $(b_1,c_3)$. $\square$

Theorem 6.2

There is a one-to-one correspondence between the incidence graph of SPBIBDs with parameters $(v,b,r,k,\lambda _1,0)$ of type $(k-1,t)$ where $0< t<k$ and distance-semiregular graphs with distance-regularized vertices of eccentricity 4.

Proof

By Lemma 6.1, $\Gamma$ is the incidence graph of a $(1+\frac{b_0b_1}{c_2}+\frac{b_0b_1b_2b_3}{c_2c_3c_4},b_0+\frac{b_0b_1b_2}{c_2c_3},b_0',b_0, c_2,0)$ SPBIBD of type $(b_1,c_3)$. Notice that $0<c_3<b_0'$ and $b_0<b_0+\frac{b_0b_1b_2}{c_2c_3}$. The result now immediately follows from Lemma 5.2. $\square$

Next, we show that if the graph $\Gamma$ from Lemma 6.1 is a bipartite distance-regularized graph, then the corresponding SPBIBD is quasi-symmetric with intersection numbers $x=0$ and $y=c_2'$.

Lemma 6.3

Let $\Gamma$ be a $(Y,Y')$-bipartite distance-regularized graph with vertices of eccentricity 4. Let $b_i, c_i; b_i', c_i' \; (0\le i \le 4)$ denote the intersection numbers of every vertex in Y and in $Y'$, respectively. Then, $\Gamma$ is the incidence graph of a $(1+\frac{b_0b_1}{c_2}+\frac{b_0b_1b_2b_3}{c_2c_3c_4},b_0+\frac{b_0b_1b_2}{c_2c_3},b_0,b_0', c_2,0)$ SPBIBD of type $(b_1,c_3)$ which is quasi-symmetric with intersection numbers $x=0$ and $y=c_2'$.

Proof

We consider the combinatorial incidence structure $\mathcal {D}=(Y, Y', \mathcal {I})$, with point set Y, block set $Y'$ and incidence $\mathcal {I}$. Observe that every bipartite distance-regularized graph with vertices of eccentricity 4 is either a bipartite distance-regular graph of diameter 4 or a distance-biregular graph with $D=D'=4$. In particular, it is distance semiregular with respect to both color partitions. Therefore, since $\Gamma$ has vertices of eccentricity 4, by Lemma 6.1, $\Gamma$ is the incidence graph of a $(1+\frac{b_0b_1}{c_2}+\frac{b_0b_1b_2b_3}{c_2c_3c_4},b_0+\frac{b_0b_1b_2}{c_2c_3},b_0',b_0, c_2,0)$ SPBIBD of type $(b_1,c_3)$. We next assert that such a 1-design is quasi-symmetric with $x=0$ and $y=c_2'$. To prove our claim, pick $B\in Y'$. We observe $\Gamma _0(B)\cup \Gamma _2(B)\cup \Gamma _4(B)=Y'$ since vertices of $Y'$ have eccentricity 4 and $\Gamma$ is bipartite. Let $B' \in \Gamma _2(B)\cup \Gamma _4(B)$. Notice that both $\Gamma _2(B)$ and $\Gamma _4(B)$ are nonempty. If $B' \in \Gamma _4(B)$ then $B\cap B'=\emptyset$. Suppose now $B' \in \Gamma _2(B)$. Since $\Gamma$ is distance-regularized we have $|\Gamma (B)\cap \Gamma (B')|=c_2'$. This shows that two intersecting blocks have the same number of points in common. Therefore, $\mathcal {D}$ is quasi-symmetric with $x=0$ and $y=c_2'$. $\square$

Finally, we prove our main result.

Theorem 6.4

There is a one-to-one correspondence between the incidence graph of quasi-symmetric SPBIBDs with parameters $(v,b,r,k,\lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $0< y\le t<k$, and bipartite distance-regularized graphs with vertices of eccentricity 4.

Proof

Let $\Gamma$ be a $(Y,Y')$-bipartite distance-regularized graphs with vertices of eccentricity 4. Let $b_i, c_i; b_i', c_i' \; (0\le i \le 4)$ denote the intersection numbers of every vertex in Y and in $Y'$, respectively. Then, by Lemma 6.3, $\Gamma$ is the incidence graph of a $(1+\frac{b_0b_1}{c_2}+\frac{b_0b_1b_2b_3}{c_2c_3c_4},b_0+\frac{b_0b_1b_2}{c_2c_3},b_0',b_0, c_2,0)$ SPBIBD of type $(b_1,c_3)$ which is quasi-symmetric with intersection numbers $x=0$ and $y=c_2'$. Notice that $0<c_2'\le c_3<b_0'$ and $b_0<b_0+\frac{b_0b_1b_2}{c_2c_3}$. The result now immediately follows from Theorem 5.6. $\square$

The classification of 2-Y-homogeneous bipartite distance-regularized graphs, which serve as the incidence graphs of quasi-symmetric special partially balanced incomplete block designs (SPBIBDs) characterized by parameters $(v,b,r,k, \lambda _1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$, is a direct consequence of Theorem 6.4 along with the outcomes from the preceding section. This classification is explicitly detailed in the subsequent theorem.

Theorem 6.5

Let $\Gamma$ be a $(Y,Y')$-bipartite distance-regularized graph. The following (i),(ii) are equivalent:

• (i) $\Gamma$ is 2-Y-homogeneous with $D=4$ and $c'_2=1$.

• (ii) $\Gamma$ is isomorphic to the subdivision graph of a complete bipartite graph $K_{n,n}=(X,\mathcal {R})$ with $n \ge 2$ and $Y=X$.

Proof

Suppose that $\Gamma$ is 2-Y-homogeneous with $D=4$ and $c'_2=1$. We notice that every vertex in $Y'$ has eccentricity 4. In fact, if $\Gamma$ is regular then $\Gamma$ is distance-regular and so, every vertex has the same eccentricity. Otherwise, by [9, Proposition 5.8] we have that $D=D'=4$. By Theorem 6.4, we therefore have that $\Gamma$ is the incidence graph of a quasi-symmetric $(v,b,r,k,\lambda _1,0)$ SPBIBD of type $(k-1,t)$ with intersection numbers $x=0$ and $y=1$. We observe that either $k\le 2$ or $r\le 2$ as otherwise, by Propositions 5.8 and 5.13, we have that $k\ge 3$ and $r\ge 3$ contradicts the fact that $\Gamma$ is 2-Y-homogeneous. Moreover, since $\Gamma$ is connected and $D=D'=4$, either $k=2$ or $r=2$. Then, by Propositions 5.7 and 5.12 we have that $\Gamma$ is isomorphic to the subdivision graph of a complete bipartite graph $K_{n,n}=(X,\mathcal {R})$ with $n \ge 2$ and $Y=X$. Conversely, if $\Gamma$ is isomorphic to the subdivision graph of a complete bipartite graph $K_{n,n}=(X,\mathcal {R})$ with $n \ge 2$ and $Y=X$, then $\Gamma$ is a $(Y,Y')$-bipartite distance-regularized graph with vertices of valency 2, $D=4$ and $c'_2=1$; see for instace [8, Theorem 2.6]. If $\Gamma$ is regular then it is isomorphic to a cycle of length 8, which is 2-Y-homogeneous with $D=4$ and $c'_2=1$. Otherwise, $\Gamma$ is distance-biregular with $k'=2$, which is 2-Y-homogeneous by [8, Theorem 4.2]. $\square$