Generalized Group–Subgroup Pair Graphs

Abstract

A regular finite graph is called a Ramanujan graph if its zeta function satisfies an analog of the Riemann Hypothesis. Such a graph has a small second eigenvalue so that it is used to construct cryptographic hash functions. Typically, explicit family of Ramanujan graphs are constructed by using Cayley graphs. In the paper, we introduce a generalization of Cayley graphs called generalized group–subgroup pair graphs, which are a generalization of group–subgroup pair graphs defined by Reyes-Bustos. We study basic properties, especially spectra of them.

1 Introduction

A k-regular finite graph is called a Ramanujan graph if its zeta function satisfies an analog of the Riemann hypothesis. This condition is equivalent to say that every nontrivial (i.e. ${}\ne \pm k$) eigenvalue of the graph is less than or equal to $2\sqrt{k-1}$. Thus the second largest eigenvalue in absolute value of a Ramanujan graph is small, and this means that it has a large isoperimetric constant (i.e. it is an expander graph), so that random walks on such a graph rapidly converge to the uniform distribution as the number of walk steps tends to infinity. Consequently, as an application to cryptography, Ramanujan graphs can be used to construct cryptographic hash functions (see Charles et al. (2009), in which hash functions are constructed from LPS graphs Lubotzky et al. (1988) and Pizer graphs (1990)).

In order to construct (a family of) Ramanujan graphs, the Cayley graphs are an important tool; a Cayley graph is a graph whose vertex set is a finite group, and the adjacency of vertices is described in terms of the multiplication of the group. In fact, most of the known explicit constructions of infinite families of Ramanujan graphs are given as Cayley graphs, and the construction is based on deep results in number theory associated with the group (for instance, the construction of the LPS graphs due to Lubotzky et al. (1988) is based on the Ramanujan–Petersson conjecture on automorphic forms).

Thus it is natural to consider the generalization of Cayley graphs to enlarge the possibility to produce Ramanujan graphs and/or expander families. Group–subgroup pair graphs (or pair graph for short) Reyes-Bustos (2016), which are defined for a triplet (G, H, S) of a finite group G, a subgroup $H\subset G$ and a suitable subset $S\subset G$, are one of such attempts. A pair graph is regular in special cases and provides interesting examples of Ramanujan graphs. However, we can construct regular pair graphs only when $[G:H]\le 2$. The purpose of this paper is to give a generalization of group–subgroup pair graphs, which can provide Ramanujan graph even when $[G:H]>2$. A generalized pair graph is a graph defined for a pair (G, H) of a group and its subgroup together with a suitable family $\mathcal {S}$ of subsets in G. We study basic properties, especially spectra of them.

Here is the brief description on the organization of the paper: In Sect. 2, we recall basic conventions on graphs. In Sect. 3, we recall the definitions of Cayley graphs and group–subgroup pair graphs, and give several examples of them. In Sect. 4, we introduce the notion of homogeneity of a graph. In Sect. 5, we give a generalization of group–subgroup pair graphs. In Sect. 6, we describe the spectra of generalized group–subgroup pair graphs.

1.1 Conventions

For a matrix A, $A^*$ is the transposed complex conjugate of A, and ${{\,\mathrm{Tr}\,}}(A)$ is the trace of A. The n by n identity matrix is denoted by $I_n$.

For a group G, we use the symbol e to indicate the identity element of G. We denote by $\chi ^\rho$ the character of a given representation $\rho$ of G: $\chi ^\rho (x)={{\,\mathrm{Tr}\,}}(\rho (x))$ for $x\in G$. The unitary dual of G (i.e. the set of all equivalence classes of unitary irreducible representations of G) is denoted by $\widehat{G}$. The dual group of G is defined to be $G^*={{\,\mathrm{Hom}\,}}(G,\mathbb {C}^{\times })$. We often identify $G^*$ with the subset

consisting of 1-dimensional representations of G via the bijection $\pi \mapsto \chi ^\pi$. When G is abelian, we have $G^*=\widehat{G}$. We denote by $\textit{\textbf{1}}$ the trivial character of G (i.e. $\textit{\textbf{1}}(x)=1$, $x\in G$).

2 Preliminaries

In what follows, a graph is always assumed to be finite, undirected and simple otherwise stated.

Let $X=(V,E)$ be a graph. The number of vertices $\left|V\right|$ and edges $\left|E\right|$ are called the order and size of the graph, respectively. We often write $x\sim y$ to indicate that two vertices x and y are adjacent, i.e. $xy\in E$. We denote by $\mathcal {N}(x)$ the neighborhood of x: . The degree $\deg (x)$ of a vertex x is the number of edges incident to x. If X is simple, then $\deg (x)$ is equal to $\left|\mathcal {N}(x)\right|$.

We call X a k-regular graph if $\deg (x)=k$ for every $x\in V$. We introduce two generalizations of this notion for later use. Suppose that V has a partition $V=V_1\sqcup \dots \sqcup V_m$.

• (1) If the degree is constant on each subset $V_i$, say $d_i$, then we call X a $(d_1,\dots ,d_m)$-regular graph.

• (2) If

  

  depends only on i and j, then we say X is a D-regular graph, where $D=(d_{ij})_{1\le i,j\le m}$. Notice that if

  $$\begin{aligned} \sum _{i=1}^m d_{ir}=\sum _{j=1}^m d_{rj}=:d_r\quad (r=1,\dots ,m), \end{aligned}$$

  then X is $(d_1,\dots ,d_m)$-regular ($\deg (x)=d_i$ for any $x\in V_i$).

Numbering the vertices, say $V=\{v_1,\dots ,v_N\}$ $(N=\left|V\right|)$, we define the adjacency matrix $\mathcal {A}=\mathcal {A}_X$ of X by

$$\begin{aligned} \mathcal {A}=(a_{ij})_{1\le i,j\le N},\quad a_{ij}={\left\{ \begin{array}{ll} 1 &{} v_i\sim v_j, \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

$\mathcal {A}$ depends on the choice of numbering of V, however, it is uniquely determined up to conjugation by permutation matrices. An eigenvalue of $\mathcal {A}$ is called an eigenvalue of the graph X. We denote by ${{\,\mathrm{Spec}\,}}(X)$ the multiset consisting of eigenvalues of X. If X is k-regular, then k is the largest eigenvalue of X, and every eigenvalue of X lies in the interval $[-k,k]$. We put

X is called a Ramanujan graph if

$$\begin{aligned} \lambda (X)\le 2\sqrt{k-1}. \end{aligned}$$

Remark 1

This condition $\lambda (X)\le 2\sqrt{k-1}$ is equivalent to the analog of the Riemann hypothesis

$$\begin{aligned} \zeta _X(q^{-s})^{-1}=0 \quad (q=k-1) \;\Longrightarrow \;{{\,\mathrm{Re}\,}}(s)=\frac{1}{2} \end{aligned}$$

for the Ihara zeta function

$$\begin{aligned} \zeta _X(u)=\prod _{[P]}(1-u^{\nu (P)})^{-1} \end{aligned}$$

of X, where [P] runs over all the “primes” in X, and $\nu (P)$ is the “length” of P. See, for example, Terras (2011) for detail.

Remark 2

It is known that the second largest eigenvalue $\lambda _1$ of X satisfies

$$\begin{aligned} \lambda _1>2\sqrt{k-1}-\frac{2\sqrt{k-1}-1}{m} \end{aligned}$$

when ${{\,\mathrm{diam}\,}}(X)\ge 2m+2\ge 4$, where ${{\,\mathrm{diam}\,}}(X)$ denotes the diameter of X Nilli (1991).

Remark 3

The notion of Ramanujan graphs is extended to non-regular graphs in several cases. For instance, a (p, q)-regular bipartite graph X is called Ramanujan bigraph if

$$\begin{aligned} \left|\sqrt{p-1}-\sqrt{q-1}\right|\le \lambda (X) \le \sqrt{p-1}+\sqrt{q-1}. \end{aligned}$$

See, for example, Feng and Li (1996), Hashimoto (1989).

Example 1

The cycle graph $C_n$ of order n is a 2-regular graph, and its eigenvalues are given by $2\cos \frac{2j\pi }{n}$ $(j=0,1,\dots ,n-1)$, which are all less than or equal to $2=2\sqrt{2-1}$. Hence $C_n$ is Ramanujan for any $n\ge 3$.

3 Cayley Graphs and Group–Subgroup Pair Graphs

We briefly recall the basics of the Cayley graphs and group–subgroup pair graphs. We refer to Fulton and Harris (1991) for basic facts on representation theory.

3.1 Cayley Graphs

Definition 1

Let G be a group and $S\subset G$ be a symmetric generating set, that is, $S^{-1}=S$ and $\left\langle S\right\rangle =G$. The Cayley graph ${{\,\mathrm{Cay}\,}}(G,S)$ is a graph whose vertex set is G and two vertices $x,y\in G$ are adjacent if and only if $y=xs$ for some $s\in S$.

Let $\mathcal {R}$ be the left regular representation of G, which is the permutation representation induced from the left translation. Explicitly, if we index the elements in G as $G=\{g_1,\dots ,g_N\}$ $(N=\left|G\right|)$, then $\mathcal {R}(g)$ $(g\in G)$ can be realized as a matrix whose (i, j)-entry is $\delta (g_i^{-1}gg_j)$, where $\delta (x)$ is 1 if $x=e$ and 0 otherwise. Then the adjacency matrix $\mathcal {A}$ of ${{\,\mathrm{Cay}\,}}(G,S)$ is given by

$$\begin{aligned} \mathcal {A}=\sum _{s\in S}\mathcal {R}(s). \end{aligned}$$

Since the irreducible decomposition of $\mathcal {R}$ is given by

$$\begin{aligned} \mathcal {R}\sim \bigoplus _{\pi \in \widehat{G}}\pi ^{\oplus \deg \pi }, \end{aligned}$$

there exists a certain unitary matrix U such that

$$\begin{aligned} U^*\mathcal {R}(g)U=\bigoplus _{\pi \in \widehat{G}}\pi (g)^{\oplus \deg \pi }. \end{aligned}$$

It follows that

$$\begin{aligned} U^*\mathcal {A}U=\bigoplus _{\pi \in \widehat{G}}\Bigl (\sum _{s\in S}\pi (s)\Bigr )^{\oplus \deg \pi }, \end{aligned}$$

and hence the characteristic polynomial of the adjacency matrix $\mathcal {A}$ is written as

$$\begin{aligned} \det (x\,I_N-\mathcal {A})=\prod _{\pi \in \widehat{G}}\det \Bigl (x\,I_{\deg \pi }-\sum _{s\in S}\pi (s)\Bigr )^{\deg \pi }. \end{aligned}$$

When G is abelian, every irreducible representation of G is 1-dimensional and we have

Example 2

Let $G=D_n=\left\langle s,t\right\rangle$ be the dihedral group of degree 2n ($s^n=t^2=e$, $tst=s^{-1}$). Take a symmetric generating subset $S=\{s,s^{-1},t\}$. Then the Cayley graph ${{\,\mathrm{Cay}\,}}(G,S)$ is a 3-regular graph which is isomorphic to the Cartesian product of the path graph $P_1$ of length 1 and the cycle graph $C_n$ of length n (Fig. 1). The following are the pictures of ${{\,\mathrm{Cay}\,}}(G,S)$ for $n=5,6,7,8$:

The eigenvalues of ${{\,\mathrm{Cay}\,}}(G,S)$ are given by

$$\begin{aligned} 2\cos \frac{2j\pi }{n}\pm 1\quad (j=0,1,\dots ,n-1). \end{aligned}$$

We see that ${{\,\mathrm{Cay}\,}}(G,S)$ is no longer Ramanujan if $2\cos \frac{2\pi }{n}+1>2\sqrt{2}$ or $n\ge 16$.

Fig. 1

${{\,\mathrm{Cay}\,}}(D_n,S)$ for $n=5,6,7,8$

3.2 Group–Subgroup Pair Graphs

Definition 2

(Reyes-Bustos (2016)) Let G be a group, H a subgroup of G and $S\subset G$ such that $S_0=S\cap H$ is symmetric (i.e. $S_0^{-1}=S_0$). The group–subgroup pair graph (or pair graph for short) $\mathcal {G}(G,H,S)$ is a graph whose vertex set is G and two vertices $x,y\in G$ are adjacent if and only if there exist $h\in H$ and $s\in S$ such that $\{x,y\}=\{h,hs\}$.

Remark 4

If $G=H=\left\langle S\right\rangle$, then $\mathcal {G}(G,G,S)={{\,\mathrm{Cay}\,}}(G,S)$. If $[G:H]=2$ and $S_0=\varnothing$, then $\mathcal {G}(G,H,S)$ is bipartite.

Example 3

If $H=\{e\}$ and $S=G\setminus \{e\}$, then $\mathcal {G}(G,H,S)$ is the star graph $K_{1,k}$ (with $\left|G\right|=k+1$). For instance, the pair graph for $G=\mathbb {Z}_8=\mathbb {Z}/8\mathbb {Z}$, $H=\{0\}$ and $S=\mathbb {Z}_8\setminus \{0\}$ is

Here we summarize several elementary facts on pair graphs (see Reyes-Bustos (2016) for the proof). Assume that H is a subgroup of G with index $k+1$ and order n. Put $N=\left|G\right|=(k+1)n$ for short. Fix a set $\{x_0=e,x_1,x_2,\dots ,x_k\}$ of representatives of the right cosets in G modulo H:

$$\begin{aligned} G=\bigsqcup _{i=0}^k V_i,\qquad V_i:=Hx_i, \end{aligned}$$

and put $S_i=Hx_i\cap S$. We also put $d_i=\left|S_i\right|$ and $d=\left|S\right|$. We denote by $\mathcal {A}$ the adjacency matrix for $\mathcal {G}(G,H,S)$, and by $\lambda _i$ $(i=0,1,\dots ,N-1)$ the eigenvalues of $\mathcal {G}(G,H,S)$ which are ordered in decreasing order: $\lambda _0\ge \lambda _1\ge \dots \ge \lambda _{N-1}$.

• We have

  $$\begin{aligned} \deg (v)={\left\{ \begin{array}{ll} d &{} v\in V_0=H, \\ d_i &{} v\in V_i\quad (i=1,\dots ,k). \end{array}\right. } \end{aligned}$$

  In particular, $\mathcal {G}(G,H,S)$ is regular if and only if $k=0$ or $k=1$ and $S_0=\varnothing$.

• $\mathcal {G}(G,H,S)$ is a D-regular graph for

  

• $\mathcal {G}(G,H,S)$ is bipartite if and only if $S_0=\emptyset$. The bipartition of G is then given by $V_0$ and $\bigcup _{i=1}^kV_i$.

• $\mathcal {G}(G,H,S)$ is connected if and only if $S_i\ne \varnothing$ for all $i\ge 1$ and $S_0\cup \bigcup _{i=1}^kS_iS_i^{-1}$ generates H (Theorem 3.3 in Reyes-Bustos (2016)).

• $\mathcal {G}(G,H,S)$ has eigenvalues (called trivial eigenvalues; see Theorem 5.1 in Reyes-Bustos (2016))

  $$\begin{aligned} \mu _{\pm }=\frac{1}{2}\Bigl (d_0\pm \bigl (d_0^2+4\sum _{i=1}^kd_i^2\bigr )^{1/2}\Bigr ). \end{aligned}$$

  $\mu _+$ is the largest eigenvalue, and it is simple if $\mathcal {G}(G,H,S)$ is connected. For any eigenvalue $\lambda$ of $\mathcal {G}(G,H,S)$ other than $\pm \lambda _0$, we have $\left|\lambda \right|<\lambda _0$.

• When $[G:H]=2$, $\mathcal {G}(G,H,S)$ is Ramanujan if $\left|S\right|\ge n+2-2\sqrt{n}$.

When the subgroup H is abelian, the eigenvalues of $\mathcal {G}(G,H,S)$ can be expressed in terms of group characters of H as follows.

Theorem 1

(Kimoto 2018, Theorem 3) If H is abelian, then the eigenvalues of $\mathcal {G}(G,H,S)$ are given by

$$\begin{aligned} \lambda _{\varphi ,\pm }=\frac{1}{2}\bigg (\sum _{h\in H_0}\varphi (h)\pm \Bigl (\Bigl (\sum _{h\in H_0}\varphi (h)\Bigr )^2+4\sum _{j=1}^k\Bigl |\sum _{h\in H_i}\varphi (h)\Bigr |^2\Bigr )^{1/2}\biggr ) \quad (\varphi \in H^*) \end{aligned}$$

and zeros whose multiplicity is at least $(k-1)n$. Here $H_i:=S_ix_i^{-1}\subset H$.

4 Homogeneity

We introduce a simple notion concerning the symmetry of a graph. Let $X=(V,E)$ be a graph. Assume that a group G acts on V. We say that X is G-homogeneous if $x\sim y$ implies $gx\sim gy$ for any $g\in G$. This is equivalent to say that G is embedded in the graph automorphism group ${{\,\mathrm{Aut}\,}}(X)$ of the graph X. We see that $\mathcal {N}(gx)=g\mathcal {N}(x)$ and hence $\deg (x)=\deg (gx)$ for any $x\in V$ and $g\in G$. In particular, if $G\curvearrowright V$ is transitive (i.e. for any $x,y\in V$, we can find $g\in G$ such that $y=gx$), then X is regular.

Remark 5

X is ${{\,\mathrm{Aut}\,}}(X)$-homogeneous.

Remark 6

A G-homogeneous graph X is vertex-transitive (i.e. for any $x,y\in V$, there exists a graph isomorphism f such that $y=f(x)$) if $G\curvearrowright V$ is transitive.

Example 4

A Cayley graph $X={{\,\mathrm{Cay}\,}}(G,S)$ is G-homogeneous by the natural left translation $(g,x)\mapsto gx$. X is $G\times G$-homogeneous via $((g_1,g_2),x)\mapsto g_1xg_2^{-1}$ if and only if S is normal or G-conjugate invariant (i.e. $gSg^{-1}=S$ for all $g\in G$) or S is a union of several conjugacy classes of G. In such a case, we have

$$\begin{aligned} \det (x\,I_N-\mathcal {A})=\prod _{\pi \in \widehat{G}} \Bigl (x-\frac{1}{\deg \pi }\sum _{s\in S}\chi ^\pi (s)\Bigr )^{(\deg \pi )^2} \end{aligned}$$

by Schur’s lemma since $\sum _{s\in S}\pi (s)$ commutes with every $\pi (g)$ $(g\in G)$ for each $\pi \in \widehat{G}$. Here $\chi ^\pi$ is the character of $\pi$.

Example 5

A pair graph $X=\mathcal {G}(G,H,S)$ is H-homogeneous.

Proposition 1

Let $X=(V,E)$ be a graph with a group action $G\curvearrowright V$ which is free (i.e. stabilizer of any $v\in V$ is trivial) and transitive. Then $X\cong {{\,\mathrm{Cay}\,}}(G,S)$ for a certain $S\subset G$.

Proof

We have $\mathcal {N}(gv)=g\mathcal {N}(v)$ for each $g\in G$ and $s\in S$. There exists $S\subset G$ such that . It is straightforward to check that $X\cong {{\,\mathrm{Cay}\,}}(G,S)$.$\square$

We roughly observe that the spectra ${{\,\mathrm{Spec}\,}}(X)$ of a graph X tends to be simple if X is equipped with a large symmetry. Pair graphs can be regarded as a class of graphs which have weakened but nontrivial symmetry (or homogeneity) compared to Cayley graphs.

In the following section, we introduce a generalization of pair graphs, which are free but non-transitive H-homogeneous graphs.

5 Generalized Group–Subgroup Pair Graph

5.1 Definition

Let G be a finite group and H its subgroup of index $k+1$. For later use, we put $N=\left|G\right|$, $n=\left|H\right|$ (hence we have $N=(k+1)n$). Fix a collection of representatives $\{x_0=e,x_1,\dots ,x_k\}$ of $H\backslash G$ and put $V_i=Hx_i$ $(i=0,1,\dots ,k)$. Let $\mathcal {S}=\{S_{ij}\}_{i,j=0}^k$ be a family of subsets in G such that

• (1) $S_{ij}\subset V_i^{-1}V_j=x_i^{-1}Hx_j$,

• (2) $e\notin S_{ij}$,

• (3) $S_{ij}^{-1}=S_{ji}$.

For two vertices $x,y\in G$, we connect these two by an edge if and only if $y=xs$ for some $s\in S_{ij}$ when $x\in V_i$ and $y\in V_j$ $(i,j=0,1,\dots ,k)$. We denote this graph by $\mathcal {G}(G,H,\mathcal {S})$, and call such a graph a generalized group–subgroup pair graph, or simply generalized pair graph. Put

$$\begin{aligned} D=\begin{pmatrix} d_{00} &{} d_{01} &{} \dots &{} d_{0k} \\ d_{10} &{} d_{11} &{} \dots &{} d_{1k} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ d_{k0} &{} d_{k1} &{} \dots &{} d_{kk} \end{pmatrix} \end{aligned}$$

with $d_{ij}=\left|S_{ij}\right|$. Notice that D is symmetric. We also put

$$\begin{aligned} d_s=\sum _{j=0}^k d_{sj}=\sum _{i=0}^k d_{is}\quad (s=0,1,\dots ,k). \end{aligned}$$

Then $\mathcal {G}(G,H,\mathcal {S})$ is a D-regular and $(d_0,d_1,\dots ,d_k)$-regular graph. Thus, if every row sum and column sum of D is equal to d, then $\mathcal {G}(G,H,\mathcal {S})$ is d-regular. By the definition, we readily see that the following lemma holds.

Lemma 1

$\mathcal {G}(G,H,\mathcal {S})$ is H-homogeneous, that is, $x\sim y$ implies $hx\sim hy$ for any $x,y\in G$ and $h\in H$.

When $k=1$ or $[G:H]=2$, H is normal and $G/H\cong \mathbb {Z}/2\mathbb {Z}$, and hence it follows that

$$\begin{aligned} S_{00},S_{11}\subset V_0,\quad S_{01},S_{10}\subset V_1. \end{aligned}$$

In this case, $\mathcal {G}(G,H,\mathcal {S})$ is $(d_0,d_1)$-biregular, and it is regular if $\left|S_{00}\right|=\left|S_{11}\right|$.

Remark 7

When $S_{ii}=\varnothing$ $(i=0,1,\dots ,k)$, then $\mathcal {G}(G,H,\mathcal {S})$ is a multi-partite graph.

5.2 Examples

Example 6

Let $X=(V,E)$ be a graph of order $k+1$ with $V=\{0,1,\dots ,k\}$, and $\mathcal {A}=(a_{ij})_{0\le i,j\le k}$ be its adjacency matrix. Take a group $G=\{x_0,x_1,\dots ,x_k\}$ of order $k+1$, and put $H=\{e\}$ and

$$\begin{aligned} S_{ij}={\left\{ \begin{array}{ll} \varnothing &{} a_{ij}=0, \\ \{x_i^{-1}x_j\} &{} a_{ij}=1. \end{array}\right. } \end{aligned}$$

Then $\mathcal {G}(G,H,\mathcal {S})\cong X$. Thus any finite graph is captured in the framework of generalized pair graphs (with trivial symmetry).

Example 7

Let G be a finite group, H its subgroup of index $k+1$ and $S\subset G$ a subset such that $S\cap H$ is symmetric. Fix a collection of representatives $\{x_0=e,x_1,\dots ,x_k\}$ of $H\backslash G$ and put $V_i=Hx_i$ $(i=0,1,\dots ,k)$. Define

$$\begin{aligned} S_{0i}=S\cap V_i,\;\;&S_{i0}=S_{0i}^{-1}\quad (i=0,1,\dots ,k),\\ S_{ij}=&\,\varnothing \quad (i\ne 0, j\ne 0). \end{aligned}$$

Then $\mathcal {G}(G,H,\mathcal {S})$ is reduced to the original group–subgroup pair graph $\mathcal {G}(G,H,S)$.

Example 8

Let $G=D_n=\left\langle s,t\right\rangle$ be the dihedral group of degree 2n. We take $H=\left\langle s\right\rangle$ and $x_0=e$, $x_1=t$. Put

$$\begin{aligned} S_{00}=\{s,s^{-1}\},\;\; S_{01}=S_{10}=\{t\},\;\; S_{11}=\{s^2,s^{-2}\}. \end{aligned}$$

Then $\mathcal {G}(G,H,\mathcal {S})$ is a $\begin{pmatrix}2 &{} 1 \\ 1 &{} 2\end{pmatrix}$-regular graph (and hence it is 3-regular). The following are the pictures of $\mathcal {G}(G,H,\mathcal {S})$ for $n=5,6,7,8$ (Fig. 2): when $n=5$, $\mathcal {G}(G,H,\mathcal {S})$ is isomorphic to the Petersen graph (the leftmost one in the picture above). These four examples are Ramanujan graphs:

$$\begin{aligned} \det (x\,I-\mathcal {A}) ={\left\{ \begin{array}{ll} (x-3)(x-1)^5(x+2)^4 &{} n=5, \\ (x-3)(x-1)x^2(x+2)^2(x^2-5)(x^2-2)^2 &{} n=6, \\ (x-3)(x-1)(x^6+2x^5-6x^4-10x^3+10x^2+11x-1)^2 &{} n=7, \\ (x-3)(x-1)(x^2-5)(x^2+2x-1)^2(x^4-4x^2+1)^2 &{} n=8 \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \lambda (X)\approx {\left\{ \begin{array}{ll} 2 &{} n=5, \\ 2.2361 &{} n=6, \\ 2.3319 &{} n=7, \\ 2.4142 &{} n=8, \end{array}\right. } \end{aligned}$$

which are less than $2\sqrt{2}\approx 2.8284$. In general, the eigenvalues of $\mathcal {G}(G,H,\mathcal {S})$ are given by

$$\begin{aligned} \cos \frac{2\pi j}{n}+\cos \frac{4\pi j}{n}\pm \sqrt{\Bigl (\cos \frac{2\pi j}{n}-\cos \frac{4\pi j}{n}\Bigr )^2+1} \quad (j=0,1,\dots ,n-1). \end{aligned}$$

$\mathcal {G}(G,H,\mathcal {S})$ is Ramanujan whenever $n\le 23$, and is not Ramanujan when $n\ge 24$.

Fig. 2

$\mathcal {G}(G,H,\mathcal {S})$ for $n=5,6,7,8$

Example 9

Let $G=D_n$ be the dihedral group of degree 2n, and we take $H=\left\langle s\right\rangle$ and $x_0=e$, $x_1=t$. Put

$$\begin{aligned} S_{00}=\{s,s^{-1}\},\; S_{01}=S_{10}=\{st,s^{-1}t\},\; S_{11}=\{s^2,s^{-2}\}. \end{aligned}$$

Then $\mathcal {G}(G,H,\mathcal {S})$ is a $\begin{pmatrix}2 &{} 2 \\ 2 &{} 2\end{pmatrix}$-regular graph (and hence it is 4-regular). The following are the pictures of $\mathcal {G}(G,H,\mathcal {S})$ for $n=5,6,7,8$ (Fig. 3): these four examples are Ramanujan graphs:

$$\begin{aligned} \det (x\,I-\mathcal {A})= {\left\{ \begin{array}{ll} x(x-4)(x^4+2x^3-4x^2-5x+5)^2 &{} n=5, \\ x^3(x-4)(x+2)^2(x^2-8)(x^2-2)^2 &{} n=6, \\ x(x-4)(x^6+2x^5-8x^4-15x^3+14x^2+28x+7)^2 &{} n=7, \\ x^3(x-4)(x+2)^2(x^2-8)(x^4-6x^2+4)^2 &{} n=8 \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \lambda (X)\approx {\left\{ \begin{array}{ll} 2.4667 &{} n=5, \\ 2.8284 &{} n=6, \\ 2.6377 &{} n=7, \\ 2.8284 &{} n=8, \end{array}\right. } \end{aligned}$$

which are less than $2\sqrt{3}\approx 3.4641$. In general, the eigenvalues of $\mathcal {G}(G,H,\mathcal {S})$ are given by

$$\begin{aligned} \cos \frac{2\pi j}{n}+\cos \frac{4\pi j}{n}\pm \sqrt{\Bigl (\cos \frac{2\pi j}{n}-\cos \frac{4\pi j}{n}\Bigr )^2+4\cos ^2\frac{2\pi j}{n}} \quad (j=0,1,\dots ,n-1). \end{aligned}$$

$\mathcal {G}(G,H,\mathcal {S})$ is Ramanujan whenever $n\le 15$, and is not Ramanujan when $n\ge 16$.

In general, when $[G:H]=2$, take $S_{00}\subset H=Hx_0$ such that $S_{00}^{-1}=S_{00}$ and $S_{01}\subset Hx_1$. We also take a nontrivial group automorphism f of H. Put $S_{11}=f(S_{00})$ and $S_{10}=S_{01}^{-1}$. Then we get a regular graph $\mathcal {G}(G,H,\mathcal {S})$.

Fig. 3

$\mathcal {G}(G,H,\mathcal {S})$ for $n=5,6,7,8$

6 Spectra of $\mathcal {G}(G,H,\mathcal {S})$

6.1 Adjacency Matrix of $\mathcal {G}(G,H,\mathcal {S})$

Let $\mathcal {A}$ be the adjacency matrix of $\mathcal {G}(G,H,\mathcal {S})$. For a concrete description of $\mathcal {A}$, we write $H=\{h_0,\dots ,h_{n-1}\}$ with $h_0=e$, and put $g_{ni+j}=h_jx_i$ for $i=0,\dots ,k$ and $j=0,\dots ,n-1$. Thus we have $G=\{g_0,g_1,\dots ,g_{N-1}\}$. Then $\mathcal {A}$ is of the form

$$\begin{aligned} \mathcal {A}=\begin{pmatrix} \mathcal {A}_{00} &{} \mathcal {A}_{01} &{} \dots &{} \mathcal {A}_{0k} \\ \mathcal {A}_{10} &{} \mathcal {A}_{11} &{} \dots &{} \mathcal {A}_{1k} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathcal {A}_{k0} &{} \mathcal {A}_{k1} &{} \dots &{} \mathcal {A}_{kk} \end{pmatrix}, \end{aligned}$$

where each block $\mathcal {A}_{pq}$ $(0\le p,q\le k)$ is given by

$$\begin{aligned} (\mathcal {A}_{pq})_{ij} ={\left\{ \begin{array}{ll} 1 &{} h_i^{-1}h_j\in H_{pq}:=x_pS_{pq}x_q^{-1}, \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

We notice that we can express each $\mathcal {A}_{pq}$ as

$$\begin{aligned} \mathcal {A}_{pq}=\sum _{s\in H_{pq}}\mathcal {R}_H(s), \end{aligned}$$

where $\mathcal {R}_H$ is the left regular representation of H.

6.2 When H is abelian

If H is abelian, then $\mathcal {R}_H$ is a direct sum of all inequivalent 1-dimensional (irreducible) representations of H, that is, there exists a certain unitary matrix U such that

$$\begin{aligned} U^*\mathcal {R}_H(h)U\sim \bigoplus _{\varphi \in H^*}\varphi (h). \end{aligned}$$

Hence

$$\begin{aligned} U^*\mathcal {A}_{pq}U=\sum _{s\in H_{pq}}\bigoplus _{\varphi \in {H^*}}\varphi (s). \end{aligned}$$

Since $\{U^*\mathcal {A}_{pq}U\}_{p,q}$ commutes with each other, we have the following theorem.

Theorem 2

Assume that H is an abelian subgroup of G. The adjacency matrix $\mathcal {A}$ of the generalized pair graph $\mathcal {G}(G,H,\mathcal {S})$ is given by

$$\begin{aligned} \det (x\,I_N-\mathcal {A})=\prod _{\varphi \in H^*}\det (x\,I_{k+1}-\mathcal {A}_\varphi ), \end{aligned}$$

where $\mathcal {A}_\varphi$ with $\varphi \in H^*$ is given by

$$\begin{aligned} \mathcal {A}_\varphi =\Bigl (\sum _{s\in H_{ij}}\varphi (s)\Bigr )_{0\le i,j\le k}. \end{aligned}$$

Remark 8

When $H=\{e\}$, we see that $H^*=\{\textit{\textbf{1}}\}$ and $\mathcal {A}_{\textit{\textbf{1}}}=\mathcal {A}$. Thus the theorem above is trivial.

Remark 9

Notice that $\mathcal {A}_{\textit{\textbf{1}}}=D$. It follows that the eigenvalues of D are also eigenvalues of $\mathcal {G}(G,H,\mathcal {S})$ if H is abelian. It is natural to ask the relation between ${{\,\mathrm{Spec}\,}}(\mathcal {A})$ and ${{\,\mathrm{Spec}\,}}(D)$ when H is non-abelian. We leave this as a future problem.

Remark 10

When $\mathcal {G}(G,H,\mathcal {S})$ is a pair graph, that is, $\mathcal {A}_{ij}=O$ if $i\ne 0$ and $j\ne 0$, we have

$$\begin{aligned} \det (x\,I_N-\mathcal {A}) =x^{(k-1)n}\det \Bigl (x^2I_n-x\,\mathcal {A}_{00}-\sum _{j=1}^k \mathcal {A}_{0j}\mathcal {A}_{j0}\Bigr ) \end{aligned}$$

without any assumption on H. If H is abelian, then Theorem 1 follows immediately from the equation above.

6.3 Petersen Extension

Let G be a group, H be a subgroup of G with index 2 and $X:={{\,\mathrm{Cay}\,}}(H,S)$ be a k-regular Cayley graph. Assume that $G=H\cup Hw$ with $w\in G$. Take a group endomorphism $\sigma \in {{\,\mathrm{End}\,}}(H)$. Notice that $X':={{\,\mathrm{Cay}\,}}(H,\sigma (S))\cong X$ if $\sigma$ is an automorphism. Put

$$\begin{aligned} S_{00}=S,\quad S_{11}=\sigma (S),\quad S_{01}=\{w\},\quad S_{10}=\{w^{-1}\}. \end{aligned}$$

Then $\widetilde{X}=\mathcal {G}(G,H,\mathcal {S})$ is a $(k+1)$-regular H-homogeneous graph. We call this the Petersen extension of ${{\,\mathrm{Cay}\,}}(H,S)$. The adjacency matrix $\widetilde{\mathcal {A}}$ of $\widetilde{X}$ is given by

$$\begin{aligned} \widetilde{\mathcal {A}}=\begin{pmatrix} \mathcal {A}&{} I_n \\ I_n &{} \mathcal {A}' \end{pmatrix}, \end{aligned}$$

where $\mathcal {A}$ and $\mathcal {A}'$ are the adjacency matrices of X and $X'$, and it follows that

$$\begin{aligned} \det (x\,I_{2n}-\widetilde{\mathcal {A}})=\det (x^2I_n-x(\mathcal {A}+\mathcal {A}')+\mathcal {A}\mathcal {A}'-I_n). \end{aligned}$$

Example 10

When $G=D_5=\left\langle s,t\right\rangle$, $H=\left\langle s\right\rangle$, $S=\{s,s^{-1}\}$, $w=t$ and $\sigma \in {{\,\mathrm{Aut}\,}}(H)$ is given by $\sigma (h)=h^2$ $(h\in H)$, the Petersen extension $\mathcal {G}(G,H,\mathcal {S})$ of ${{\,\mathrm{Cay}\,}}(H,S)$ is the Petersen graph (Fig. 4).

Remark 11

If $\sigma$ is the identity map of H (i.e. $X'=X$), then the Petersen extension $\mathcal {G}(G,H,\mathcal {S})$ of ${{\,\mathrm{Cay}\,}}(H,S)$ is just a Cartesian product of ${{\,\mathrm{Cay}\,}}(H,S)$ and the path graph .

In general, it is not true that the Petersen extension $\widetilde{X}$ of $X={{\,\mathrm{Cay}\,}}(H,S)$ is Ramanujan when X is Ramanujan. Thus we propose the following problem.

Fig. 4

${{\,\mathrm{Cay}\,}}(H,S)$ and its Petersen extension $\mathcal {G}(G,H,\mathcal {S})$

Problem 1

Characterize the quintuple $(G,H,S,w,\sigma )$ such that both ${{\,\mathrm{Cay}\,}}(H,S)$ and its Petersen extension with w and $\sigma$ are Ramanujan.

6.3.1 Examples: Dihedral case

We look at the case where $G=D_n=\left\langle s,t\right\rangle$, $H=\left\langle s\right\rangle$ and $w=t$, for instance. In this case, an endomorphism $\sigma$ of H is given by $\sigma (h)=h^l$ for certain $l\in \mathbb {Z}$, and $\sigma \in {{\,\mathrm{Aut}\,}}(H)$ if and only if $\gcd (n,l)=1$. We also notice that $wSw^{-1}=tSt=S$ for any symmetric generating subset S of H.

Let $X_{n,l}:=\mathcal {G}(G,H,\mathcal {S})$ be the Petersen extension of ${{\,\mathrm{Cay}\,}}(H,S)$ defined by w and $\sigma :H\ni h\mapsto h^l\in H$. Then, the family $\mathcal {S}$ is given by

For each character $\varphi \in H^*$, define

$$\begin{aligned} \alpha _\varphi :=\sum _{s\in S}\varphi (s),\quad \beta _\varphi :=\sum _{s\in S}\varphi (s^l). \end{aligned}$$

By Theorem 2, we see that

$$\begin{aligned} \det (x\,I_{2n}-\mathcal {A})=\prod _{\varphi \in H^*}\det (x\,I_2-\mathcal {A}_\varphi ),\qquad \mathcal {A}_\varphi =\begin{pmatrix} \alpha _\varphi &{} 1 \\ 1 &{} \beta _\varphi \end{pmatrix}, \end{aligned}$$

where $\mathcal {A}$ is the adjacency matrix of $X_{n,l}$. Hence the eigenvalues of $X_{n,l}$ are given by

$$\begin{aligned} \frac{\alpha _\varphi +\beta _\varphi \pm \sqrt{(\alpha _\varphi -\beta _\varphi )^2+4}}{2} \quad (\varphi \in H^*). \end{aligned}$$

Example 11

If $n\ge 3$ and $S=\{s,s^{-1}\}$, then

$$\begin{aligned} \alpha _\varphi&=e^{\frac{2\pi ij}{n}}+e^{-\frac{2\pi ij}{n}}=2\cos \frac{2\pi j}{n}, \\ \beta _\varphi&=e^{\frac{2l\pi ij}{n}}+e^{-\frac{2l\pi ij}{n}}=2\cos \frac{2l\pi j}{n} \end{aligned}$$

for $\varphi \in H^*$ given by $\varphi (s)=e^{\frac{2\pi ij}{n}}$. Thus the eigenvalues of $X_{n,l}$ are calculated as

$$\begin{aligned} \cos \frac{2\pi j}{n}+\cos \frac{2l\pi j}{n}\pm \sqrt{\Bigl (\cos \frac{2\pi j}{n}-\cos \frac{2l\pi j}{n}\Bigr )^2+1} \quad (j=0,1,\dots ,n-1). \end{aligned}$$

We can numerically check that

• (1) if $n\le 53$ and $n\ne 48$, then there exists l such that $X_{n,l}$ is Ramanujan,

• (2) if $n\ge 54$ or $n=48$, then $X_{n,l}$ is not Ramanujan for any choice of l.

When n is odd and $\gcd (n,l)=1$ (i.e. $\sigma \in {{\,\mathrm{Aut}\,}}(H)$), then we see that

• (1) if $n\le 53$ and $n\ne 45$, then there exists l such that $X_{n,l}$ is Ramanujan,

• (2) if $n\ge 55$ or $n=45$, then $X_{n,l}$ is not Ramanujan for any choice of l.

Example 12

If $n=2m\ge 4$ is even and $S=\{s,s^m,s^{-1}\}$, then

$$\begin{aligned} \alpha _\varphi&=e^{\frac{2\pi ij}{n}}+e^{\frac{2m\pi ij}{n}}+e^{-\frac{2\pi ij}{n}}=(-1)^j+2\cos \frac{2\pi j}{n}, \\ \beta _\varphi&=e^{\frac{2l\pi ij}{n}}+e^{\frac{2lm\pi ij}{n}}+e^{-\frac{2l\pi ij}{n}}=(-1)^{lj}+2\cos \frac{2l\pi j}{n} \end{aligned}$$

for $\varphi \in H^*$ given by $\varphi (s)=e^{\frac{2\pi ij}{n}}$. We can numerically check that

• (1) if $m\le 29$ ($n\le 58$), then there exists l such that $X_{n,l}$ is Ramanujan,

• (2) if $m\ge 30$ ($n\ge 60$), then $X_{n,l}$ is not Ramanujan for any choice of l.

Example 13

If $n\ge 5$ and $S=\{s,s^2,s^{-1},s^{-2}\}$, then

$$\begin{aligned} \alpha _\varphi&=e^{\frac{2\pi ij}{n}}+e^{\frac{4\pi ij}{n}}+e^{-\frac{2\pi ij}{n}}+e^{-\frac{4\pi ij}{n}}=2\cos \frac{2\pi j}{n}+2\cos \frac{4\pi j}{n}, \\ \beta _\varphi&=e^{\frac{2l\pi ij}{n}}+e^{\frac{4l\pi ij}{n}}+e^{-\frac{2l\pi ij}{n}}+e^{-\frac{4l\pi ij}{n}}=2\cos \frac{2l\pi j}{n}+2\cos \frac{4l\pi j}{n} \end{aligned}$$

for $\varphi \in H^*$ given by $\varphi (s)=e^{\frac{2\pi ij}{n}}$. We can numerically check that

• (1) if $n\le 33$, then there exists l such that $X_{n,l}$ is Ramanujan,

• (2) if $n\ge 34$, then $X_{n,l}$ is not Ramanujan for any choice of l.

Remark 12

In the paper, we discuss the construction of graphs when a finite group G and its subgroup H are given. It would be also interesting to consider the situation where finite groups G, H and an epimorphism $p:G\twoheadrightarrow H$ are given (i.e. H is a quotient group of G).