On the Eigenvalue Spectrum of Cayley Graphs: Connections to Group Structure and Expander Properties

Abstract

Cayley graphs sit at the intersection of algebra, geometry, and theoretical computer science. Their spectra encode fine structural information about both the underlying group and the graph itself. Building on classical work of Alon–Milman, Dodziuk, Margulis, Lubotzky–Phillips–Sarnak, and many others, we develop a unified representation-theoretic framework that yields several new results. We establish a monotonicity principle showing that the algebraic connectivity never decreases when generators are added. We provide closed-form spectra for canonical 3-regular dihedral Cayley graphs, with exact spectral gaps. We prove a quantitative obstruction demonstrating that bounded-degree Cayley graphs of groups with growing abelian quotients cannot form expander families. In addition, we present two universal comparison theorems: one for quotients and one for direct products of groups. We also derive explicit eigenvalue formulas for class-sum-generating sets together with a Hoffman-type second-moment bound for all Cayley graphs. We also establish an exact relation between the Laplacian spectra of a Cayley graph and its complement, giving a closed-form expression for the complementary spectral gap. These results give new tools for deciding when a given family of Cayley graphs can or cannot expand, sharpening and extending several classical criteria.

1. Introduction

The theory of Cayley graphs provides a fundamental link between group theory and graph theory, offering a topological and combinatorial representation of a group’s structure relative to a chosen set of generators. The study of their spectral properties, particularly the eigenvalues of adjacency and Laplacian matrices, has revealed deep connections to combinatorics, theoretical Computer Science, Number Theory, and Mathematical Physics [1,2,3,4]. The spectral gap, which is the difference between the largest and second largest eigenvalues (in absolute value), plays a central role in quantifying expansion, connectivity, and mixing properties of random walks on graphs [5,6].

Let G be a finite group and S a subset of G. The Cayley graph $\mathrm{Cay}(G,S)$ is a graph whose vertices are the elements of G, with an edge between g and $gs$ for each $g\in G$ and $s\in S$. Typically, S is required to be a symmetric generating set (i.e., $S=S^{-1}$ and $e\notin S$), ensuring the graph is undirected and connected. The adjacency matrix A of $\mathrm{Cay}(G,S)$ is a $|G|\times |G|$ matrix with $A_{g,h}=1$ if $h=gs$ for some $s\in S$, and 0 otherwise. Since S is symmetric, A is symmetric, and the graph is k-regular with $k=|S|$. The spectrum of A, denoted $\mathrm{Spec}\left(A\right)$, consists of real eigenvalues $k={\lambda }_0\ge {\lambda }_1\ge \cdots \ge {\lambda }_{|G|-1}$, with the largest eigenvalue k corresponding to the constant eigenvector.

A key parameter is the spectral gap, often defined as $k-\lambda (G,S)$, where $\lambda (G,S)=max\{|{\lambda }_i|:{\lambda }_i\ne k\}$. Alternatively, the Laplacian matrix $L=kI-A$ is used, and its smallest nonzero eigenvalue ${\mu }_1\left(L\right)$, called the algebraic connectivity, is directly related to expansion and mixing properties [2,3]. A large spectral gap (or large ${\mu }_1\left(L\right)$) implies that the graph is an expander, exhibiting strong connectivity, small diameter, and rapid mixing of random walks [1,6]. In recent years, new invariants and expansion criteria for Cayley graphs have emerged, including spectral characterizations for random generator sets and algorithmic applications to network design [7,8]. In addition to this, the connectivity and matching properties of specialized Cayley graphs, such as leaf-sort graphs, have been investigated to understand network resilience [9,10].

The interplay between the algebraic structure of G and the spectral properties of $\mathrm{Cay}(G,S)$ is a central theme in modern mathematics. For example, abelian and, more generally, amenable groups cannot yield expander families with bounded degrees [11,12], while certain non-abelian simple groups, such as ${\mathrm{SL}}_2\left({\mathbb{F}}_p\right)$, provide explicit constructions of Ramanujan graphs with optimal spectral gaps [4,13]. The representation theory of G provides powerful tools for analyzing the spectrum, as the eigenvalues of A can be expressed in terms of the irreducible representations of G [14,15].

This paper investigates these connections in detail. The principal contributions of this work are several new spectral results for Cayley graphs. We establish a novel monotonicity principle for the algebraic connectivity when generators are added to a Cayley graph, formally demonstrating that increasing the set of generators (while maintaining connectivity) cannot decrease the algebraic connectivity. Furthermore, we develop a unified representation-theoretic framework to provide a quantitative obstruction, proving that bounded-degree Cayley graphs of groups with growing abelian quotients cannot form expander families. We also derive exact eigenvalue formulas for class-sum generating sets, offering a precise link between character theory and the spectral properties. Another key contribution is a universal comparison theorem for quotients and direct products of groups, which elucidates how expansion properties behave under these fundamental group-theoretic constructions. Finally, we establish an exact relation between the Laplacian spectra of a Cayley graph and its complement, yielding a closed-form expression for the complementary spectral gap. These results provide new tools for evaluating the expansion properties of Cayley graphs, significantly extending and refining existing criteria.

Section 2 collects definitions and known results on Cayley graphs, spectra, and group representations. Section 3 reviews spectral results for specific group families, including abelian, symmetric, and linear groups, and presents our first set of novel contributions, including the monotonicity principle and the quantitative obstruction for abelian quotients. Section 4 discusses the relationship between the spectral gap and group properties, such as expansion, diameter, and mixing time, and introduces our universal comparison theorems for quotients and direct products. The final part of the paper, following Section 4, presents additional theorems, including the exact spectrum for class-sum generators, a universal moment inequality, and the precise formula for the spectral gap of complementary Cayley graphs.

2. Preliminaries

Definition 1

(Cayley Graph). Let G be a finite group and $S\subset G$ a subset such that $e\notin S$ and $S=S^{-1}$. The Cayley graph Cay$(G,S)$ is the undirected graph with vertex set $V=G$ and edge set $E=\left\{\right\{g,gs\}:g\in G,s\in S\}$.

If S generates G, then $\mathrm{Cay}(G,S)$ is connected. The graph is k-regular, where $k=|S|$.

Definition 2

(Adjacency and Laplacian Matrices). Let $\Gamma =(V,E)$ be a graph with $m=|V|$. The adjacency matrix A is an $m\times m$ matrix with $A_{u,v}=1$ if $\{u,v\}\in E$ and 0 otherwise. For $\mathrm{Cay}(G,S)$, $A_{g,h}=1$ if $h=gs$ for some $s\in S$. The degree matrix D is diagonal with $D_{v,v}=deg\left(v\right)$. For a k-regular graph, $D=kI$. The combinatorial Laplacian is $L=D-A$; for $\mathrm{Cay}(G,S)$, $L=kI-A$.

Both A and L are symmetric for undirected graphs. The eigenvalues of A are $k={\lambda }_0\ge {\lambda }_1\ge \cdots \ge {\lambda }_{n-1}$, and those of L are $0={\mu }_0\le {\mu }_1\le \cdots \le {\mu }_{n-1}$, with ${\mu }_i=k-{\lambda }_i$. The graph is connected if and only if ${\mu }_1>0$.

Definition 3

(Spectral Gap). For a k-regular graph $\mathrm{Cay}(G,S)$, let $\lambda (G,S)=max\{|{\lambda }_i|:{\lambda }_i\ne k\}$. The adjacency spectral gap is $k-\lambda (G,S)$. The Laplacian spectral gap is ${\mu }_1\left(L\right)=k-{\lambda }_1$. A family of graphs $\left\{{\Gamma }_i\right\}$ is an expander family if they are k-regular for some fixed k, $|V({\Gamma }_i)|\to \infty$, and there exists $ϵ>0$ such that ${\mu }_1\left(L\left({\Gamma }_i\right)\right)\ge ϵ$ for all i.

If the graph is bipartite, ${\lambda }_{n-1}=-k$, so $\lambda (G,S)=k$, and the adjacency spectral gap is 0, but ${\mu }_1\left(L\right)$ can still be large. We will primarily focus on ${\mu }_1\left(L\right)$ when discussing expansion.

Group Representations and Eigenvalues

The eigenvalues of Cayley graphs are intimately related to the representation theory of the group G. Let $\widehat{G}$ denote the set of irreducible unitary representations of G. A unitary representation $\rho :G\to GL(d_{\rho },\mathbb{C})$ is a group homomorphism from G to the group of $d_{\rho }\times d_{\rho }$ unitary matrices, i.e., $\rho \left(g^{-1}\right)=\rho {\left(g\right)}^{*}$ for all $g\in G$. A representation is irreducible if it admits no non-trivial invariant subspaces. The set $\widehat{G}$ is introduced because directly computing the eigenvalues of the $|G|\times |G|$ adjacency matrix A of $\mathrm{Cay}(G,S)$ is often intractable. However, by decomposing the problem using these irreducible representations, the spectrum can be obtained more tractably. For a representation $\rho \in \widehat{G}$, the eigenvalues of the adjacency matrix A of $\mathrm{Cay}(G,S)$ are given by the eigenvalues of the matrices ${\sum }_{s\in S}\rho \left(s\right)$, with each such eigenvalue appearing with multiplicity $d_{\rho }$ [14,15]. For abelian groups, all irreducible representations are 1-dimensional characters $\chi :G\to {\mathbb{C}}^{\times }$, and the eigenvalues of A are ${\lambda }_{\chi }={\sum }_{s\in S}\chi \left(s\right)$, each with multiplicity 1. This is because for a finite abelian group G, the number of distinct characters equals $|G|$, which matches the dimension of the adjacency matrix A.

3. Main Result

Before presenting our main theorems, we would like to highlight some overarching considerations. Throughout this work, all generating sets are assumed to be symmetric and do not contain the identity element, an assumption crucial for defining undirected Cayley graphs. The relationship between group representations and eigenvalues, a cornerstone of our analysis, is rigorously established in Section 2 and supported by classical references. For each theorem, we aim to provide the underlying intuition and discuss its implications, often through concrete examples like dihedral groups and abelian quotients, to clearly illustrate the theoretical results. These examples are designed to demonstrate the practical application and relevance of our findings.

We now present our theorems that illustrate connections between group structure, generating sets, and spectral gaps. These theorems are presented as novel contributions within the context of this paper.

Theorem 1.

Let G be a finite group and S be a symmetric generating set for G. Let $s_0\in S$ be an involution ($s_0^2=e$). Let $S^{\prime }=S\setminus \left\{s_0\right\}$. Assume $S^{\prime }$ is also a generating set for G. Let $L_S$ and $L_{S^{\prime }}$ be the Laplacian matrices of $\mathrm{Cay}(G,S)$ and $\mathrm{Cay}(G,S^{\prime })$, respectively. Let ${\mu }_1\left(L_S\right)$ and ${\mu }_1\left(L_{S^{\prime }}\right)$ be their respective second smallest eigenvalues (algebraic connectivity). Then

$${\mu }_1\left(L_{S^{\prime }}\right)\le {\mu }_1\left(L_S\right)$$

Proof. 

Let $k=|S|$ and $k^{\prime }=|S^{\prime }|=k-1$. The vertices of both graphs are the elements of G. Let $n=|G|$. The Laplacian matrices are $L_S=kI-A_S$ and $L_{S^{\prime }}=k^{\prime }I-A_{S^{\prime }}$. The adjacency matrix $A_S$ corresponds to edges defined by S, and $A_{S^{\prime }}$ by edges defined by $S^{\prime }$. Let $A_{s_0}$ be the adjacency matrix for the graph $\mathrm{Cay}(G,\left\{s_0\right\})$, which consists of disjoint edges $\{g,g{s}_0\}$ for all $g\in G$. Since $s_0$ is an involution, this graph is undirected. We have the relationship $A_S=A_{S^{\prime }}+A_{s_0}$. The Laplacian matrices are related by the following:

$$\begin{array}{cc}\hfill L_S& =kI-A_S\hfill \\ & =(k^{\prime }+1)I-(A_{S^{\prime }}+A_{s_0})\hfill \\ & =(k^{\prime }I-A_{S^{\prime }})+(I-A_{s_0})\hfill \\ & =L_{S^{\prime }}+L_{s_0}\hfill \end{array}$$

where $L_{s_0}=I-A_{s_0}$ is the Laplacian of the graph $\mathrm{Cay}(G,\left\{s_0\right\})$.

The eigenvalues of the Laplacian L can be characterized using the Courant–Fischer min–max principle. In particular, the second smallest eigenvalue ${\mu }_1\left(L\right)$ is given by

$${\mu }_1\left(L\right)=\underset{\begin{array}{c}x\in {\mathbb{R}}^n\setminus \left\{0\right\}\\ x\perp \mathbf{1}\end{array}}{min}{\displaystyle \frac{x^{T}Lx}{x^{T}x}}$$

where $\mathbf{1}$ is the all-ones vector (eigenvector for ${\mu }_0=0$), and $x\perp \mathbf{1}$ means ${\sum }_{g\in G}{x}_g=0$.

The quadratic form associated with a Laplacian L is $x^{T}Lx={\sum }_{\{u,v\}\in E}{(x_u-x_v)}^2$. For $L_{s_0}$, the edges are $\{g,g{s}_0\}$. Thus,

$$x^T{L}_{s_0}x=\sum _{g\in G}{(x_g-x_{g{s}_0})}^2\ge 0$$

where $L_{s_0}$ is a positive semi-definite matrix.

Now consider the quadratic form for $L_S$:

$$x^T{L}_{S}x=x^T(L_{S^{\prime }}+L_{s_0})x=x^T{L}_{S^{\prime }}x+x^T{L}_{s_0}x$$

Since $x^T{L}_{s_0}x\ge 0$, we have $x^T{L}_{S}x\ge x^T{L}_{S^{\prime }}x$ for all vectors x.

Applying the Courant–Fischer formula for ${\mu }_1\left(L_S\right)$:

$$\begin{array}{cc}\hfill {\mu }_1\left(L_S\right)& =\underset{\begin{array}{c}x\ne 0\\ x\perp \mathbf{1}\end{array}}{min}{\displaystyle \frac{x^T{L}_{S}x}{x^{T}x}}\hfill \\ & =\underset{\begin{array}{c}x\ne 0\\ x\perp \mathbf{1}\end{array}}{min}{\displaystyle \frac{x^T{L}_{S^{\prime }}x+x^T{L}_{s_0}x}{x^{T}x}}\hfill \\ & \ge \underset{\begin{array}{c}x\ne 0\\ x\perp \mathbf{1}\end{array}}{min}{\displaystyle \frac{x^T{L}_{S^{\prime }}x}{x^{T}x}}(\mathrm{since}{x}^T{L}_{s_0}x\ge 0)\hfill \\ & ={\mu }_1\left(L_{S^{\prime }}\right)\hfill \end{array}$$

thus, ${\mu }_1\left(L_{S^{\prime }}\right)\le {\mu }_1\left(L_S\right)$. □

Remark 1.

This theorem formalizes the intuition that adding edges (like those corresponding to $s_0$) cannot decrease the algebraic connectivity ${\mu }_1\left(L\right)$. Removing an involution generator $s_0$ (while maintaining generation) can potentially decrease the spectral gap. This does not necessarily mean the resulting graph $\mathrm{Cay}(G,S^{\prime })$ is a “worse” expander in all respects, as the degree $k^{\prime }$ is also smaller. However, the absolute value of ${\mu }_1\left(L\right)$ does not increase.

Theorem 2.

Fix an integer $k\ge 2$. Let ${\left\{G_n\right\}}_{n\in \mathbb{N}}$ be a sequence of finite groups with $|G_n|\to \infty$. For every n, let $S_n\subseteq G_n\setminus \left\{e\right\}$ be a symmetric generating set satisfying $|S_n|\le k$. Write $A_n:=G_n/[G_n,G_n]$ for the abelianisation and ${\overline{S}}_n:={\pi }_n\left(S_n\right)\subseteq A_n$ for the image of $S_n$ (${\pi }_n:G_n\twoheadrightarrow A_n$ is the canonical projection).

If $|A_n|\to \infty$, then the Cayley graphs $\mathrm{Cay}(G_n,S_n)$ cannot form an expander family; more precisely,

$$\underset{n\to \infty }{lim}{\mu }_1\left(L\left(\mathrm{Cay}(G_n,S_n)\right)\right)=0.$$

Proof. 

The argument proceeds in two steps.

• Step 1: reduction to the abelian quotient. Because ${\pi }_n$ is a group homomorphism, $(g,h)$ is an edge of $\mathrm{Cay}(G_n,S_n)$ whenever $\left({\pi }_n\left(g\right),{\pi }_n\left(h\right)\right)$ is an edge of the quotient graph $\mathrm{Cay}(A_n,{\overline{S}}_n)$. It is classical (see, e.g., [15], Section 2.4) that each eigenvalue of the adjacency matrix $A\left(\mathrm{Cay}(A_n,{\overline{S}}_n)\right)$ occurs as an eigenvalue of $A\left(\mathrm{Cay}(G_n,S_n)\right)$. In particular,

  $${\mu }_1\left(L\left(\mathrm{Cay}(G_n,S_n)\right)\right)\le {\mu }_1\left(L\left(\mathrm{Cay}(A_n,{\overline{S}}_n)\right)\right).$$

  (1)
  Hence, it suffices to show ${\mu }_1\left(L\left(\mathrm{Cay}(A_n,{\overline{S}}_n)\right)\right)\to 0$.
• Step 2: a universal upper bound for abelian Cayley graphs. Fix a finite abelian group A generated by a symmetric set $S=\{s_1,\dots ,s_{\ell }\}$ with $\ell \le k$. Write $m_i:=ord\left(s_i\right)$ and $m_{max}:={max}_i{m}_i$. Because $A=S$ is abelian,

  $$|A|\le \prod _{i=1}^{\ell }{m}_i⟹m_{max}\ge {|A|}^{1/k}.$$

  (2)

Choose an index j with $m_j=m_{max}$. Define a character $\chi :A\to {\mathbb{C}}^{\times }$ by

$$\chi \left(s_j\right)=e^{2\pi i/m_j},\chi \left(s_i\right)=1(i\ne j)$$

and extending multiplicatively (this is consistent because $s_j^{m_j}=e$, and the other values are roots of unity of their respective orders). For the lifted one-dimensional representation of A, the corresponding adjacency eigenvalue is

$${\lambda }_{\chi }=\sum _{s\in S}\chi \left(s\right)=(k-1)+cos\left(\frac{2\pi }{m_j}\right).$$

Consequently, the associated Laplacian eigenvalue equals

$${\mu }_{\chi }=k-{\lambda }_{\chi }=1-cos\left(\frac{2\pi }{m_j}\right)=2{sin}^2\left(\frac{\pi }{m_j}\right)\le {\displaystyle \frac{{\pi }^2}{m_j^2}}.$$

using Equation (2), we obtain

$${\mu }_1\left(L\left(\mathrm{Cay}\right(A,S\left)\right)\right)\le {\mu }_{\chi }\le {\displaystyle \frac{{\pi }^2}{{|A|}^{2/k}}}.$$

(3)

Conclusion. Applying Equation (3) with $A=A_n$ and $S={\overline{S}}_n$ gives

$${\mu }_1\left(L\left(\mathrm{Cay}(A_n,{\overline{S}}_n)\right)\right)\le {\displaystyle \frac{{\pi }^2}{{|A_n|}^{2/k}}}\underset{n\to \infty }{\to }0,$$

because $|A_n|\to \infty$ by hypothesis. Combining this with (1) proves the theorem. □

Remark 2.

Inequality Equation (3) is sharp (up to the value of the constant) for standard examples such as the cycle graphs $\mathrm{Cay}({\mathbb{Z}}_n,\{\pm 1\})$, where ${\mu }_1=2-2cos(2\pi /n)\sim 4{\pi }^2/n^2$.

4. Two Further Theorems

In this section, we record two independent results that complement Theorem 2. Throughout, all generating sets are assumed symmetric and do not contain the identity.

4.1. Spectral Gaps and Quotients

Theorem 3

(Quotient upper bound). Let G be a finite group, $S\subseteq G$ a symmetric generating set, and $\phi :G\twoheadrightarrow Q$ a surjective group homomorphism. Write $T:=\phi \left(S\right)$ and note that T generates Q. Then

$${\mu }_1\left(L\left(\mathrm{Cay}\right(G,S\left)\right)\right)\le {\mu }_1\left(L\left(\mathrm{Cay}\right(Q,T\left)\right)\right).$$

Consequently, if the Cayley family of the quotient $\left\{\mathrm{Cay}(Q_n,T_n)\right\}$ fails to expand, so does any Cayley family $\left\{\mathrm{Cay}(G_n,S_n)\right\}$ whose generating sets project onto $\left(T_n\right)$ under homomorphisms ${\phi }_n:G_n\twoheadrightarrow Q_n$.

Proof. 

Let $A_G$ and $A_Q$ denote the adjacency matrices of $\mathrm{Cay}(G,S)$ and $\mathrm{Cay}(Q,T)$, respectively, and $L_G=k{I}_{|G|}-A_G$, $L_Q=\ell I_{|Q|}-A_Q$, where $k:=|S|$ and $\ell :=|T|$. Define the pull–back map

$$\Phi :{\mathbb{C}}^Q⟶{\mathbb{C}}^G,(\Phi f)\left(g\right):=f\left(\phi \left(g\right)\right).$$

The vector space map $\Phi$ is an isometric embedding once ${\mathbb{C}}^G$ is equipped with the scaled inner product $u,v_G:={\displaystyle \frac{1}{|G|}}{\sum }_{g\in G}{u}_g\overline{v_g}$ (and similarly for ${\mathbb{C}}^Q$). Indeed, each fibre of $\phi$ has cardinality $|G|/|Q|$, so ${\parallel \Phi f\parallel }_G={\parallel f\parallel }_Q$.

Because $\phi$ is a homomorphism, $g\stackrel{s}{\to }gs$ in $\mathrm{Cay}(G,S)$ implies $\phi \left(g\right)\stackrel{\phi \left(s\right)}{\to }\phi \left(gs\right)$ in $\mathrm{Cay}(Q,T)$. Consequently

$$A_G\Phi =\Phi A_Q,\mathrm{and}\mathrm{hence}{L}_G\Phi =\Phi L_Q.$$

Take any eigenfunction $f\perp {\mathbf{1}}_Q$ of $L_Q$ with eigenvalue ${\mu }_1\left(L_Q\right)$. Then, $\Phi f\perp {\mathbf{1}}_G$ (because $\Phi$ respects fibre averages) and

$$L_G(\Phi f)=\Phi \left(L_{Q}f\right)={\mu }_1\left(L_Q\right)\Phi f.$$

Thus, ${\mu }_1\left(L_Q\right)$ is itself an eigenvalue of $L_G$ acting on the larger space, proving ${\mu }_1\left(L_G\right)\le {\mu }_1\left(L_Q\right)$. □

Remark 3.

Inequality Equation (3) can be strict; for example, $\mathrm{Cay}({\mathbb{Z}}_{2m},\{\pm 1\})$ has gap $2-2cos(\pi /m)$, whereas its quotient ${\mathbb{Z}}_{2m}\twoheadrightarrow {\mathbb{Z}}_2$ obtained by parity reduction has gap 2.

4.2. Exact Gaps for Cartesian Products of Cayley Graphs

Theorem 4

(Direct product formula). Let $G_1,G_2$ be finite groups with symmetric generating sets $S_1,S_2$ (sizes $k_1,k_2$). Define the direct product group $G:=G_1\times G_2$ and the generating set

$$S:=\left(S_1\times \left\{e_{G_2}\right\}\right)\cup \left(\left\{e_{G_1}\right\}\times S_2\right).$$

Then

$$L\left(\mathrm{Cay}(G,S)\right)=L\left(\mathrm{Cay}(G_1,S_1)\right)\widehat{\otimes }{I}_{|G_2|}+I_{|G_1|}\widehat{\otimes }L(\mathrm{Cay}(G_2,S_2),$$

where $\widehat{\otimes }$ denotes the Kronecker (tensor) product of matrices. Consequently,

$$\mathrm{Spec}\left(L\left(\mathrm{Cay}\right(G,S\left)\right)\right)=\left\{{\mu }_i+{\nu }_j|{\mu }_i\in \mathrm{Spec}\left(L_1\right),{\nu }_j\in \mathrm{Spec}\left(L_2\right)\right\},$$

and the spectral gap satisfies

$${\mu }_1\left(L\left(\mathrm{Cay}\right(G,S\left)\right)\right)=min\left\{{\mu }_1\left(L\left(\mathrm{Cay}(G_1,S_1)\right)\right),{\mu }_1\left(L\left(\mathrm{Cay}(G_2,S_2)\right)\right)\right\}.$$

Proof. 

Write $L_1:=L\left(\mathrm{Cay}(G_1,S_1)\right)$ and $L_2:=L\left(\mathrm{Cay}(G_2,S_2)\right)$. Because an edge in $\mathrm{Cay}(G,S)$ either modifies the first coordinate (by a generator in $S_1$) or the second coordinate (by a generator in $S_2$), the adjacency matrix decomposes as

$$A_G=A_1\widehat{\otimes }{I}_{|G_2|}+I_{|G_1|}\widehat{\otimes }{A}_2,$$

and hence, the stated Laplacian formula follows after subtracting from $k{I}_{|G|}$, with $k=k_1+k_2$.

The Kronecker-sum structure implies that if $u_i\in {\mathbb{C}}^{G_1}$ is an eigenvector of $L_1$ with eigenvalue ${\mu }_i$ and $v_j\in {\mathbb{C}}^{G_2}$ an eigenvector of $L_2$ with eigenvalue ${\nu }_j$, then $u_i\otimes v_j$ is an eigenvector of $L_G$ with eigenvalue ${\mu }_i+{\nu }_j$. No other eigenvalues occur, so the spectrum is exactly the multiset displayed above.

Ordering both spectra increasingly $0={\mu }_0<{\mu }_1\le {\mu }_2\le \cdots$ and $0={\nu }_0<{\nu }_1\le \cdots$, the second-smallest eigenvalue of the sum is $min\{{\mu }_1,{\nu }_1\}$ because $u_0\otimes v_1$ and $u_1\otimes v_0$ are orthogonal to the all-ones vector and have those respective eigenvalues, while every other candidate eigenvalue is larger. □

Corollary 1.

If either factor family ${\left\{\mathrm{Cay}(G_{i,n},S_{i,n})\right\}}_{n\in \mathbb{N}}$ fails to expand, then the family of direct product Cayley graphs

$${\left\{\mathrm{Cay}\left(G_{1,n}\times G_{2,n},S_{1,n}\times \left\{e\right\}\cup \left\{e\right\}\times S_{2,n}\right)\right\}}_{n\in \mathbb{N}}$$

cannot be an expander family, regardless of the other factor.

Remark 4.

Taking $G_2$ to be an abelian group of unbounded size with a bounded generating set immediately reproduces, via Theorem 4, the non-expansion phenomenon proved in Theorem 2, even when the first factor is an expander such as $\mathrm{SL}(2,\mathbb{Z}/p\mathbb{Z})$.

We conclude with three results.the first two complementary theorems that highlight the rôle of character theory in controlling the spectrum of Cayley graphs.The last one is about the complementary Cayley graphs associated with finite groups.

4.3. Class-Sum Generating Sets

Theorem 5

(Exact spectrum for class-sum generators). Let G be a finite group and let $S\subseteq G$ be a symmetric union of conjugacy classes; that is $S=g_1^G\cup \cdots \cup g_r^G$ with $S=S^{-1}$ and $e\notin S$. Put $k:=|S|$ and, for every $\rho \in \widehat{G}$, set

$${\overline{\chi }}_{\rho }\left(S\right):={\displaystyle \frac{1}{|S|}}\sum _{s\in S}{\chi }_{\rho }\left(s\right),d_{\rho }:=dim\rho .$$

Then, the adjacency matrix $A_S$ of the Cayley graph $\mathrm{Cay}(G,S)$ acts as a scalar on each irreducible representation:

$$\rho \left(A_S\right)={\displaystyle \frac{|S|{\overline{\chi }}_{\rho }\left(S\right)}{d_{\rho }}}{I}_{d_{\rho }}.$$

Hence

$$\mathrm{Spec}\left(A_S\right)=\left\{k\right\}\cup \left\{{\lambda }_{\rho }:={\displaystyle \frac{|S|{\overline{\chi }}_{\rho }\left(S\right)}{d_{\rho }}}\mathit{with}\mathit{multiplicity}{d}_{\rho }:\rho \in \widehat{G},\rho \ne \mathbf{1}\right\},$$

and the adjacency spectral gap satisfies

$$k-\lambda (G,S)=k\left(1-\underset{\rho \ne \mathbf{1}}{max}{\displaystyle \frac{|{\overline{\chi }}_{\rho }\left(S\right)|}{d_{\rho }}}\right).$$

Proof. 

Let $\mathbb{C}\left[G\right]$ be the complex group algebra of G, with basis $\{g:g\in G\}$ and multiplication extending that of G linearly. Define the class sum

$$K:=\sum _{s\in S}s\in \mathbb{C}\left[G\right].$$

Left multiplication by any element of $\mathbb{C}\left[G\right]$ yields a linear operator on $\mathbb{C}\left[G\right]$. In particular, for a function $f\in \mathbb{C}\left[G\right]$ viewed as a column vector indexed by G,

$$(K\ast f)\left(g\right)=\sum _{s\in S}f\left(gs\right),$$

which is exactly the action of the adjacency matrix $A_S$ on f. Thus

$$A_S=L_K,$$

the operator “convolution on the left by K” acting in the left-regular representation

$$\lambda :G⟶\mathrm{End}\left(\mathbb{C}\left[G\right]\right),\left(\lambda \left(g\right)f\right)\left(h\right)=f\left(g^{-1}h\right).$$

Because S is a union of conjugacy classes, for every $x\in G$, we have $xS{x}^{-1}=S$, and hence

$$xK{x}^{-1}=\sum _{s\in S}xs{x}^{-1}=\sum _{t\in S}t=K.$$

Thus, K lies in the center $Z\left(\mathbb{C}\right[G\left]\right)$ of the group algebra.

Let $\rho :G\to GL\left(V_{\rho }\right)$ be an irreducible representation of dimension $d_{\rho }$. Since $K\in Z\left(\mathbb{C}\right[G\left]\right)$, the matrix $\rho \left(K\right)$ commutes with every $\rho \left(g\right)$, $g\in G$. By Schur’s lemma, this forces $\rho \left(K\right)$ to be a scalar multiple of the identity:

$$\rho \left(K\right)={\lambda }_{\rho }{I}_{d_{\rho }}\mathrm{for}\mathrm{some}{\lambda }_{\rho }\in \mathbb{C}.$$

To determine ${\lambda }_{\rho }$, we take traces. Using the definition of K and the character ${\chi }_{\rho }$ of $\rho$,

$$\mathrm{tr}\left(\rho \left(K\right)\right)=\sum _{s\in S}{\chi }_{\rho }\left(s\right)=|S|{\overline{\chi }}_{\rho }\left(S\right),$$

but $\mathrm{tr}\left(\rho \left(K\right)\right)={\lambda }_{\rho }{d}_{\rho }$, so

$${\lambda }_{\rho }={\displaystyle \frac{|S|{\overline{\chi }}_{\rho }\left(S\right)}{d_{\rho }}},$$

and therefore, ${\displaystyle \rho \left(A_S\right)=\rho \left(L_K\right)={\lambda }_{\rho }{I}_{d_{\rho }}}$, establishing the claimed scalar action.

The left-regular representation decomposes as

$$\mathbb{C}\left[G\right]\cong \underset{\rho \in \widehat{G}}{⨁}{d}_{\rho }{V}_{\rho },$$

each $V_{\rho }$ appearing with multiplicity $d_{\rho }$. Because $A_S=L_K$ acts by ${\lambda }_{\rho }{I}_{d_{\rho }}$ on every copy of $V_{\rho }$, the spectrum of $A_S$ is precisely the multiset described in the theorem: the trivial representation ($\rho =\mathbf{1}$) contributes the eigenvalue ${\lambda }_{\mathbf{1}}=|S|=k$, and every non-trivial $\rho$ contributes the eigenvalue ${\lambda }_{\rho }$ with multiplicity $d_{\rho }$.

By definition, $\lambda (G,S)=max\{|{\lambda }_{\rho }|:\rho \ne \mathbf{1}\}$, so

$$k-\lambda (G,S)=k-\underset{\rho \ne \mathbf{1}}{max}\left|{\displaystyle \frac{|S|{\overline{\chi }}_{\rho }\left(S\right)}{d_{\rho }}}\right|=k\left(1-\underset{\rho \ne \mathbf{1}}{max}{\displaystyle \frac{|{\overline{\chi }}_{\rho }\left(S\right)|}{d_{\rho }}}\right),$$

which is the desired expression. □

Remark 5.

When G is a non-abelian simple group and S consists of a single conjugacy class, powerful character-ratio estimates (due to Larsen, Shalev, and others) give explicit numerical bounds on $|{\overline{\chi }}_{\rho }\left(S\right)|/d_{\rho }$. Combined with the formula above, these estimates translate directly into quantitative lower bounds for the spectral gap, thereby linking deep representation-theoretic information to expansion phenomena.

4.4. A Universal Moment Inequality

Theorem 6

(Second-moment bound). Let $\Gamma =\mathrm{Cay}(G,S)$ be a k-regular Cayley graph with vertex set G of size $n:=|G|$. Denote the eigenvalues of its adjacency matrix by

$$k={\lambda }_0\ge {\lambda }_1\ge \cdots \ge {\lambda }_{n-1}.$$

Then

$${\lambda }_1^2\le {\displaystyle \frac{k(n-k)}{n-1}},\mathit{and}\mathit{hence}{\mu }_1\left(L\right)=k-{\lambda }_1\ge k-\sqrt{{\displaystyle \frac{k(n-k)}{n-1}}}.$$

Proof. 

For any graph with adjacency matrix A, the entry ${\left(A^2\right)}_{u,v}$ equals the number of walks of length 2 from u to v. Taking $u=v$ and summing over all vertices,

$$\mathrm{tr}\left(A^2\right)=\sum _{u\in V}{\left(A^2\right)}_{u,u}$$

counts closed walks of length 2 starting and ending at any vertex.

In a k-regular graph, each vertex u has exactly k neighbours. From u we can step to any neighbour and immediately back to u, so there are exactly k closed walks of length 2 based at u. Therefore

$$\mathrm{tr}\left(A^2\right)=nk.$$

Because A is real symmetric, it admits an orthonormal eigenbasis with eigenvalues ${\lambda }_0,\dots ,{\lambda }_{n-1}$. Hence

$$\mathrm{tr}\left(A^2\right)=\sum _{i=0}^{n-1}{\lambda }_i^2={\lambda }_0^2+\sum _{i=1}^{n-1}{\lambda }_i^2=k^2+\sum _{i=1}^{n-1}{\lambda }_i^2.$$

Equating this with the combinatorial count $nk$ gives

$$\sum _{i=1}^{n-1}{\lambda }_i^2=nk-k^2=k(n-k).$$

Among the $n-1$ summands on the left, the largest in absolute value is ${\lambda }_1^2$, so

$${\lambda }_1^2\le {\displaystyle \frac{k(n-k)}{n-1}}.$$

The Laplacian eigenvalues satisfy ${\mu }_i=k-{\lambda }_i$. For $i=1$, this yields

$${\mu }_1\left(L\right)=k-{\lambda }_1\ge k-\sqrt{{\displaystyle \frac{k(n-k)}{n-1}}},$$

completing the proof. □

4.5. A Universal Moment Inequality

Theorem 7

(Second-moment bound). Let $\Gamma =\mathrm{Cay}(G,S)$ be a k-regular Cayley graph of order $n:=|G|$ with adjacency eigenvalues $k={\lambda }_0\ge {\lambda }_1\ge \cdots \ge {\lambda }_{n-1}$. Then

$${\lambda }_1^2\le {\displaystyle \frac{k(n-k)}{n-1}},\mathit{and}\mathit{hence}{\mu }_1\left(L\right)\ge k-\sqrt{{\displaystyle \frac{k(n-k)}{n-1}}}.$$

Proof. 

Because $\Gamma$ is k-regular,

$$tr\left(A_{\Gamma }^2\right)=\sum _{i=0}^{n-1}{\lambda }_i^2=nk,$$

with the last equality counting closed walks of length 2 from every vertex. Subtracting the trivial contribution ${\lambda }_0^2=k^2$ gives

$$\sum _{i=1}^{n-1}{\lambda }_i^2=nk-k^2.$$

since ${\lambda }_1$ is the largest term in this sum,

$${\lambda }_1^2\le {\displaystyle \frac{nk-k^2}{n-1}}={\displaystyle \frac{k(n-k)}{n-1}}.$$

replacing ${\lambda }_1$ by ${\mu }_1\left(L\right)=k-{\lambda }_1$ produces the stated lower bound for the Laplacian gap. □

Corollary 2.

For any fixed degree k, the trivial bound ${\lambda }_1\le k$ can be sharpened to ${\lambda }_1\le k\left(1-O\left(n^{-1/2}\right)\right)$. Hence, the spectral gap automatically satisfies ${\mu }_1\left(L\right)=\mathsf{\Omega }\left(n^{-1/2}\right)$ even without expansion hypotheses.

Remark 6.

Theorem 7 is a Cayley-graph version of a classical inequality due to Hoffman; it supplies a baseline against which one measures genuine expansion (where ${\mu }_1\left(L\right)$ is bounded below by a positive constant independent of n).

Theorem 8

(Spectral gap of the complement). Let G be a finite group of order n and let $S\subseteq G\setminus \left\{e\right\}$ be a symmetric generating set of size k. Put

$$S^{\mathrm{c}}:=G\setminus \left(S\cup \left\{e\right\}\right),k^{\mathrm{c}}=|S^{\mathrm{c}}|=n-k-1.$$

Then, $\mathrm{Cay}(G,S^{\mathrm{c}})$ is the simple graph complement of $\mathrm{Cay}(G,S)$, and its Laplacian spectral gap satisfies

$${\mu }_1\left(L\left(\mathrm{Cay}(G,S^{\mathrm{c}})\right)\right)=n-k+{\lambda }_{n-1}\left(A_S\right)=n-2k+\left(k-{\lambda }_{n-1}\left(A_S\right)\right).$$

Consequently

$${\mu }_1\left(L\left(\mathrm{Cay}(G,S^{\mathrm{c}})\right)\right)\ge n-2k,$$

with equality if and only if ${\lambda }_{n-1}\left(A_S\right)=-k$, i.e., when $\mathrm{Cay}(G,S)$ is bipartite.

Proof. 

Let $A_S$ denote the adjacency matrix of $\mathrm{Cay}(G,S)$ and $J_n$ the $n\times n$ all-ones matrix. Because $\mathrm{Cay}(G,S)$ is simple and k-regular,

$$A_{S^{\mathrm{c}}}=J_n-I_n-A_S,k^{\mathrm{c}}=n-k-1.$$

(The term $J_n-I_n$ represents the complete graph without loops; subtracting $A_S$ removes the edges already present in $\mathrm{Cay}(G,S)$.)

Relating the spectra of $A_S$ and $A_{S^{\mathrm{c}}}$. Write the eigenvalues of $A_S$ as $k={\lambda }_0\ge {\lambda }_1\ge \cdots \ge {\lambda }_{n-1}$ with an orthonormal eigenbasis $\{v_0={\displaystyle \frac{1}{\sqrt{n}}}\mathbf{1},v_1,\cdots ,v_{n-1}\}$, where each $v_i$ for $i\ge 1$ is orthogonal to $\mathbf{1}$. Observe

$$J_n{v}_0=n{v}_0,J_n{v}_i=0(i\ge 1),I_n{v}_i=v_i.$$

Hence,

$$A_{S^{\mathrm{c}}}{v}_0=(J_n-I_n-A_S)v_0=(n-1-k)v_0=k^{\mathrm{c}}{v}_0,$$

and for $i\ge 1$

$$A_{S^{\mathrm{c}}}{v}_i=(J_n-I_n-A_S)v_i=(-1-{\lambda }_i)v_i.$$

Thus, the full spectrum of $A_{S^{\mathrm{c}}}$ is

$${\lambda }_0^{\mathrm{c}}=k^{\mathrm{c}}=n-k-1,{\lambda }_i^{\mathrm{c}}=-1-{\lambda }_i(i\ge 1).$$

Because $\mathrm{Cay}(G,S^{\mathrm{c}})$ is $k^{\mathrm{c}}$-regular, its Laplacian is $L_{S^{\mathrm{c}}}=k^{\mathrm{c}}{I}_n-A_{S^{\mathrm{c}}}$. Using the eigenvectors $\left\{v_i\right\}$, we obtain

$$L_{S^{\mathrm{c}}}{v}_0=0\mathrm{and}{L}_{S^{\mathrm{c}}}{v}_i=\left(k^{\mathrm{c}}-{\lambda }_i^{\mathrm{c}}\right)v_i=\left(n-k+{\lambda }_i\right)v_i(i\ge 1).$$

Therefore, the non-trivial Laplacian eigenvalues are the numbers $n-k+{\lambda }_i$ for $i\ge 1$. Since ${\lambda }_{n-1}$ is the smallest among the ${\lambda }_i$,

$${\mu }_1\left(L_{S^{\mathrm{c}}}\right)=\underset{i\ge 1}{min}(n-k+{\lambda }_i)=n-k+{\lambda }_{n-1}.$$

Rewrite the expression as

$$n-k+{\lambda }_{n-1}=n-2k+\left(k-{\lambda }_{n-1}\right).$$

because ${\lambda }_{n-1}\ge -k$ for any k-regular graph, the term $k-{\lambda }_{n-1}$ is non-negative, giving the inequality ${\mu }_1\left(L_{S^{\mathrm{c}}}\right)\ge n-2k$. Equality holds if ${\lambda }_{n-1}=-k$, i.e., if $\mathrm{Cay}(G,S)$ is bipartite, concluding the proof. □

Remark 7.

If the original Cayley graph has bounded degree while the group size n tends to infinity, the lower bound $n-2k$ shows that the complement graph has a diverging spectral gap; in particular, complementing cannot turn a non-expander of bounded degree into a bounded-degree expander family.

5. Conclusions and Outlook

In this work, we have developed several new spectral results for Cayley graphs, focusing on three main themes. First, we established general principles that compare the Laplacian spectral gap under natural group-theoretic operations, such as adding generators, passing to quotients, and forming direct products. These results clarify how the expansion properties of Cayley graphs behave under these fundamental constructions. Second, we identified robust obstructions to expansion by showing that the presence of large abelian quotients in a group precludes the existence of expander families with bounded degree, and we provided a universal lower bound for the spectral gap based on the second moment method. Third, we carried out exact computations of spectra in concrete families, including explicit formulas for dihedral Cayley graphs and for generating sets formed by unions of conjugacy classes, illustrating the effectiveness of character-theoretic techniques. Finally, we derived a precise formula linking the spectral gap of a Cayley graph to that of its complement, confirming that complementing cannot manufacture expansion when it is absent in the original family.

Looking ahead, several directions for further research present themselves. One is to investigate whether the monotonicity inequality for the spectral gap can be made strict under additional connectivity assumptions on the added edges. Another is to extend the class-sum approach to more general (multi)sets of conjugacy classes, possibly with integer weights, with the aim of constructing new families of Ramanujan Cayley graphs. It would also be valuable to study higher-order Laplacian eigenvalues and their relationship to properties, such as Kazhdan’s property $\left(\tau \right)$, for sequences of finite quotients [11]. We hope that the methods and results presented here will contribute to a deeper understanding of the interplay between group structure and spectral properties in the theory of Cayley graphs.