1 Introduction

Let Γ be a distance-regular graph of diameter d. (For notation and definitions related to distance-regular graphs, see [4].) For primitive strongly regular graphs (the case d=2) it is known that Γ 2(γ), the subgraph of Γ induced by the vertices at distance 2 from a given vertex γ (also called the second subconstituent of γ), is connected. See [4, p. 86] or [3, Proposition 9.3.1] for an eigenvalue proof due to Haemers or [6] for a combinatorial proof of this fact. The properties of the second subconstituents of strongly regular graphs have been studied by many authors [3–6, 8] and their connectedness is a very important property of strongly regular graphs (cf. [3, Sect. 12.6]).

During the GAC5 conference [2] (see also [1]), Andries Brouwer asked whether this could be generalized to a statement that for general Γ and suitable t, the subgraph Γ ≥t (γ) is connected, where Γ ≥t (γ) is the subgraph of Γ induced by the vertices of distance at least t to γ. In this paper, we show that one can take for t the position where the standard sequence corresponding to the second largest eigenvalue changes sign.

2 Results

Let the distance-regular graph Γ have intersection array {b 0,…,b d−1;c 1,…,c d } and put k=b 0, a i =k−b i −c i as usual. Define the tridiagonal matrices

$$L_i = \left[\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 & b_0 \\ c_1 & a_1 & b_1 \\ & c_2 & a_2 & b_2 \\ & & \cdot & \cdot & \cdot \\ & & & \cdot & \cdot & \cdot \\ & & & & c_{i-1} & a_{i-1} & b_{i-1} \\ & & & & & c_i & a_i \end{array} \right] $$

and

$$M_i = \left[\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} a_i & b_i \\ c_{i+1} & a_{i+1} & b_{i+1} \\ & \cdot & \cdot & \cdot \\ & & \cdot & \cdot & \cdot \\ & & & c_{d-1} & a_{d-1} & b_{d-1} \\ & & & & c_d & a_d \end{array} \right]. $$

Let ρ i be the largest eigenvalue of L i , and let σ i be the largest eigenvalue of M i . For γ∈V(Γ), Γ ≤i (γ) denotes the subgraph of Γ induced by the vertices of distance at most i from γ and Γ ≥i (γ) denotes the subgraph of Γ induced by the vertices at distance at least i from γ. Then ρ i is the largest eigenvalue of the subgraphs Γ ≤i (γ) of Γ, and σ i is the largest eigenvalue of the subgraphs Γ ≥i (γ) of Γ.

Proposition 1

(i) The ρ i are increasing: ρ i <ρ j when 0≤i<j≤d.

(ii) The σ i are decreasing: σ i >σ j when 0≤i<j≤d.

(iii) ρ i ≤σ d−i for 0≤i≤d.

Proof

The Perron–Frobenius Theorem (see [4, Sect. 3.1], [7, Sect. 2.6] or [8, Sect. 8.8]) tells us that the largest eigenvalue of a graph is at least as large as the largest eigenvalue of a subgraph, and this gives (iii) because Γ ≤i (δ) is a subgraph of Γ ≥d−i (γ) when d(γ,δ)=d. That theorem moreover says that the inequality is strict for a proper subgraph of a connected graph, and this gives (i) because all Γ ≤i (δ) are connected, and also (ii) because each connected component of Γ ≥j (γ) is a proper subgraph of some connected component of Γ ≥i (γ). □

Let L=L d =M 0. Let Γ have the d+1 distinct eigenvalues k=θ 0>θ 1>⋯>θ d . Then the θ i are the eigenvalues of the matrix L. The standard sequence u=(u 0,…,u d )⊤ of an eigenvalue θ of Γ is the corresponding right eigenvector of L, so that Lu=θu, normalized such that u 0=1. From now on, let u denote the standard sequence of the second largest eigenvalue θ 1 of Γ. It is known that it has precisely one sign change (see [4, Chap. 4]). In particular, u d <0.

Proposition 2

Let 1≤t≤d−1.

  1. (i)

    If u t =0, then ρ t−1=θ 1=σ t+1.

  2. (ii)

    If u t >0, then ρ t−1<θ 1<σ t+1.

  3. (iii)

    If u t <0, then ρ t−1>θ 1>σ t+1.’

Proof

Since the ρ i are increasing, and the σ i are decreasing, when proving (ii) we may assume that t is the largest index for which u t >0, and when proving (iii) we may assume that t is the smallest index for which u t <0.

Let x=(u 0,…,u t−1)⊤ and y=(u t+1,…,u d )⊤. Now v:=L t−1 x−θ 1 x=(0,…,0,−b t−1 u t )⊤, and the last component of v is zero, negative, resp. positive in the three cases of the proposition. Let z ⊤ be a positive left eigenvector of L t−1 for its largest eigenvalue ρ t−1. We see that (ρ t−1−θ 1)z ⊤ x=z ⊤ v is zero, negative, positive in the three cases, while z ⊤ x>0. This proves the left-hand inequalities.

Similarly, w:=M t+1 y−θ 1 y=(−c t+1 u t ,0,…,0)⊤, and the first component of w is zero, negative, resp. positive in the three cases of the proposition. Let z ⊤ be a positive left eigenvector of M t+1 for its largest eigenvalue σ t+1. We see that (σ t+1−θ 1)z ⊤ y=z ⊤ w is zero, negative, positive in the three cases, while z ⊤ y<0. This proves the right-hand inequalities. □

Since ρ i ≤σ j when i+j≤d, and ρ i <σ j when i+j<d, it follows that t>d/2 when u t <0, and t≥d/2 when u t ≤0. See also [10] for some related results.

Theorem 3

Let Γ be a distance-regular graph of diameter d>0. Let u=(u 0,…,u d )⊤ be the standard sequence corresponding to the second largest eigenvalue of Γ. If u t−1>0, then Γ ≥t (γ) is connected for each vertex γ of Γ.

Proof

Let H:=Γ ≥t (γ), and let A H be its adjacency matrix. Let C be a connected component of H. Let z be a positive eigenvector of M t (indexed by {t,…,d}) for its largest eigenvalue σ t . The vector y defined by y δ =z d(γ,δ) for δ∈C, and y δ =0 for δ∈V(H)∖C is an eigenvector of H with eigenvalue σ t >θ 1. By eigenvalue interlacing (see [4, Sect. 3.3], [3, Sect. 2.5], [7, Chap. 5] or [8, Chap. 9]), H can have at most a single eigenvalue larger than θ 1, so H has only one connected component. □

3 Final remarks and open problems

The result stated in the introduction that the second subconstituent of a primitive strongly regular graph is connected, can be deduced from our work. The standard sequence u=(u 0,u 1,u 2) corresponding to the second largest eigenvalue θ 1 of connected strongly regular graph has a sign change at u 1 (as u 0=1,u 1=θ 1/k and u 2 is negative) unless θ 1≤0. This happens if and only if the connected strongly regular graph is a complete multipartite graph.

We remark that there are many distance-regular graphs Γ of diameter d≥3 with the property that the subgraph induced by Γ d (γ) is not connected for any γ∈V(Γ). If Γ is a distance-regular graph with k d >1 and a d =0, then Γ d (γ) will be disconnected for any γ∈V(Γ). Distance-regular graphs having this property include antipodal r-covers with r≥3 and bipartite distance-regular graphs with k d ≠1, but there are also primitive examples. The Patterson graph [4, p. 410] (for d=4), the Livingstone graph [4, p. 406] (for d=4) and the Biggs–Smith graph [4, p. 403] (for d=7) all have k d >1 and a d =0. The Coxeter graph [4, p. 382] has d=4, k d =6 and a d =1 (and thus Γ 4(γ) is isomorphic to 3K 2). The Sylvester graph [4, p. 394] has d=3, k d =10 and a d =1 (and thus Γ 3(γ) is isomorphic to 5K 2). The Odd graph Γ=O d+1 [4, p. 259] is the graph whose vertex set is formed by all d-subsets of a set with 2d+1 elements, where two d-subsets are adjacent if and only if they are disjoint. This graph is distance-regular with degree d+1, diameter d and \(a_{d}=\lceil \frac{d+1}{2}\rceil\). For d≥3, one can check easily that the subgraph induced by Γ d (γ) is disconnected for any γ∈V(Γ).

If d∈{3,4}, then our results imply that the induced subgraph on Γ d−1(γ)∪Γ d (γ) is connected for any γ∈V(Γ), unless d=4 and Γ is an antipodal r-cover for r≥3. All the known primitive distance-regular graphs that we have checked have Γ d−1(γ)∪Γ d (γ) connected. Is this statement true for all primitive distance-regular graphs? We showed it for diameter up to 4. If d=2s+1, then it is not clear when Γ ≥s+1(γ) is disconnected.

The Johnson graph J(n,d) has as vertices the subsets of order d of a set with n elements with two such subsets being adjacent if their intersection has size d−1. It is well known that J(n,d) is a distance-regular graph of diameter d and degree d(n−d) having b j =(d−j)(n−d−j) and c j =j 2 for each 0≤j≤d (see [4, Sect. 9.1]). For every γ∈V(J(n,d)), the induced subgraph Γ d (γ) is connected as is isomorphic to J(n−d,d). When n>d 2, we can deduce that Γ d (γ) is connected using our previous results. The standard sequence corresponding to θ 1=(d−1)(n−d−1)−1=b 1−1 is \(u_{i}=1-\frac{in}{d(n-d)}\) for 0≤i≤d (see [4, Proposition 4.4.9]). It follows easily that if n divides d(n−d), then the standard sequence has a zero u t , where \(t=\frac{d(n-d)}{n}\). If n does not divide d(n−d), then u t−1>0 and u t <0, when \(t=\lceil \frac{d(n-d)}{n} \rceil\). Thus, when n>d 2, then u d−1>0 and u d <0 which implies Γ d (γ) is connected for every γ∈V(J(n,d)). However, when n=4t−3, d=2t−2, we obtain u t−1>0 and u t <0. Also, when n=4t−1, d=2t−1, we obtain u t−1>0 and u t <0.

The Hamming graph H(d,q) has as vertex set the Cartesian product Q d, where Q is a set of q elements. Two vertices of H(d,q) are adjacent if they differ in exactly one position. It is well known that H(d,q) is a distance-regular graph of diameter d and degree d(q−1) having b j =(d−j)(q−1), c j =j for each 0≤j≤d (see [4, Sect. 9.2]). For every γ∈V(H(d,q)), the induced subgraph Γ d (γ) is connected as is isomorphic to H(d,q−1). When q>d, we can use our previous results to deduce that Γ d (γ) is connected. The standard sequence corresponding to θ 1=dq−d−q=b 1−1 is \(u_{i}=1-\frac{iq}{d(q-1)}\) for 0≤i≤d (see [4, Proposition 4.4.9]). It follows that if q divides d, then the standard sequence has a zero u t , where \(t=\frac{d(q-1)}{q}\). If q does not divide d, then u t−1>0 and u t <0, when \(t=\lceil \frac{d(q-1)}{q} \rceil\). Thus, when q>d, then u d−1>0 and u d <0 which implies Γ d (γ) is connected for every γ∈V(H(d,q)). However, when q=2, d=2t−1, then u t−1>0 and u t <0.

Motivated by the previous results, we call a distance-regular graph Γ of diameter d a generalized Shilla graph if the standard sequence corresponding to the second largest eigenvalue of Γ contains a zero. When d=3, this notion is the same as the notion of Shilla graph introduced by Koolen and Park in [9]. The previous examples show that many distance-regular graphs such as Hamming graphs or Johnson graphs are generalized Shilla graphs. It would be interesting to determine what other distance-regular graphs are generalized Shilla graphs. A related problem would be to classify all the distance-regular graphs of diameter d=3 such that Γ d (γ) is disconnected for some γ∈V(Γ). Such graphs would have to satisfy the condition u 2≤0. The case u 2=0 is particularly interesting and these graphs will have \(\theta_{1}=a_{3}=\frac{a_{1}+\sqrt{a_{1}^{2}+4k}}{2}\) (see Koolen and Park [9] for more details on such graphs). Another problem worth investigating is the relation between the index t, where the standard sequence corresponding to θ 1 changes its sign, and the index s such that k s =max0≤i≤d k i . If θ 1<k/2, then one can show that u t−1≥0 and u t <0 imply that b t−2≥k−θ 1>k/2 and c t+1>k−θ 1>k/2. This shows that b t−3>c t−2 and c t+2>b t+1 and hence k t−3<k t−2 and k t+1>k t+2. Thus, when θ 1<k/2, the parameter s satisfies t−2≤s≤t+1. It would be interesting to determine how far apart can these parameters be in general.