Let $S_n$ and $A_{n}$ denote the symmetric and alternating group on the set $\{1,.., n\},$ respectively. In this paper we are interested in the second largest eigenvalue $λ_{2}(Γ)$ of the Cayley graph $Γ=Cay(G,H)$ over $G=S_{n}$ or $A_{n}$ for certain connecting sets $H.$
Let $1<k\leq n$ and denote the set of all $k$-cycles in $S_{n}$ by $C(n,k).$ For $H=C(n,n)$ we prove that $λ_{2}(Γ)=(n-2)!$ (when $n$ is even) and $λ_{2}(Γ)=2(n-3)!$ (when $n$ is odd). Further, for $H=C(n,n-1)$ we have $λ_{2}( Γ)=3(n-3)(n-5)!$ (when $n$ is even) and $λ_{2}(Γ)=2(n-2)(n-5) !$ (when $n$ is odd). The case $H=C(n,3)$ has been considered in X. Huang and Q. Huang, The second largest eigenvalue of some Cayley graphs on alternating groups, J. Algebraic Combinatorics} 50(2019), $99-111$.
Let $1\leq r<k<n$ and let $C(n,k;r) \subseteq C(n,k)$ be set of all $k$-cycles in $S_{n}$ which move all the points in the set $\{1,2,..., r\}.$ That is to say, $g=(i_{1},i_{2}... i_{k})(i_{k+1})\dots(i_{n})\in C(n,k;r)$ if and only if $\{1,2,..., r\}\subset \{i_{1},i_{2},..., i_{k}\}.$
Our main result concerns $λ_{2}( Γ)$, where $Γ=Cay(G,H)$ with $H=C(n,k;r)$ with $1\leq r<k<n$ when $G=S_{n}$ if $k$ is even and $G=A_{n}$ if $k$ is odd. Here we observe that $$λ_{2}( Γ)\geq (k-2)! {n-r \choose k-r} \frac{1}{n-r} \big((k-1)(n-k) - \frac{(k-r-1)(k-r)}{n-r-1}\big).$$ We show that this bound is sharp in the special case $k=r+1$ , giving $λ_{2}(Γ)=r!(n-r-1)$. The cases with $H=C(n,3;1)$ and $H=C(n,3;2)$ were considered earlier in the same paper of X. Huang and Q. Huang.
We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.
If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called \emph{H-integral}. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are \emph{Gaussian integers}, then the mixed graph is called \emph{Gaussian integral}. For any finite group $Γ$, we characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(Γ, S)$ is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if it is Gaussian integral.
Let $Γ$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(Γ,T)$ be a Cayley graph of $Γ$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $G$ in terms of the second eigenvalues of certain subgraphs of $G$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$ and we determine the second eigenvalues of a majority of the connected normal Cayley g…
In 2020, Siemons and Zalesski [On the second eigenvalue of some Cayley graphs of the symmetric group. {\it arXiv preprint arXiv:2012.12460}, 2020] determined the second eigenvalue of the Cayley graph $Γ_{n,k} = \operatorname{Cay}(\operatorname{Sym}(n), C(n,k))$ for $k = 0$ and $k=1$, where $C(n,k)$ is the conjugacy class of $(n-k)$-cycles. In this paper, it is proved that for any $n\geq 3$ and $k\in \mathbb{N}$ relatively small compared to $n$, the second eigenvalue of $Γ_{n,k}$ is the eigenvalue afforded by the irreducible character of $\operatorname{Sym}(n)$ that corresponds to the partition…
Let $A_n$ denote the alternating group of degree $n$ with $n\geq 3$. The alternating group graph $AG_n$, extended alternating group graph $EAG_n$ and complete alternating group graph $CAG_n$ are the Cayley graphs $\mathrm{Cay}(A_n,T_1)$, $\mathrm{Cay}(A_n,T_2)$ and $\mathrm{Cay}(A_n,T_3)$, respectively, where $T_1=\{(1,2,i),(1,i,2)\mid 3\leq i\leq n\}$, $T_2=\{(1,i,j),(1,j,i)\mid 2\leq i<j\leq n\}$ and $T_3=\{(i,j,k),(i,k,j)\mid 1\leq i<j<k\leq n\}$. In this paper, we determine the second largest eigenvalues of $AG_n$, $EAG_n$ and $CAG_n$.
Researchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite Abelian group G whose eigenvalues are algebraic integers in a given number field K. And we succeed in finding a method to do so by proving Theorem 1. Also, the number of such Cayley digraphs is computed.
In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph G_{n}. We also obtain distance Laplacian eigenvalues and distance Laplacian energy of G_{n}.
A mixed graph is called \emph{second kind hermitian integral}(or \emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. Let $Γ$ be an abelian group. We characterize the set $S$ for which a mixed Cayley graph $\text{Cay}(Γ, S)$ is HS-integral. We also show that a mixed Cayley graph is Eisenstein integral if and only if it is HS-integral.
A mixed graph is said to be HS-\emph{integral} if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(Γ, S)$ is HS-integral for any finite group $Γ$. We further show that a normal mixed Cayley graph is HS-integral if and only if it is Eisenstein integral. This paper generalizes the results of [M. Kadyan, B. Bhattacharjya. HS-integral and Eisenstein integral mix…
Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property fo…
We express the discrete Ricci curvature of a graph as the minimal eigenvalue of a family of matrices, one for each vertex of a graph whose entries depend on the local adjaciency structure of the graph. Using this method we compute or bound the Ricci curvature of Cayley graphs of finite Coxeter groups and affine Weyl groups. As an application we obtain an isoperimetric inequality that holds for all Cayley graphs of finite Coxeter groups.
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…
A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $Γ$ be an abelian group. The \textit{mixed Cayley graph} $Cay(Γ,S)$ is a mixed graph on the vertex set $Γ$ and edge set $\{ (a,b): b-a\in S \}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(Γ,S)$ over abelian group in terms of its symbol set $S$.
We obtain a new bound connecting the first non--trivial eigenvalue of the Laplace operator of a graph and the diameter of the graph, which is effective for graphs with small diameter or for graphs, having the number of maximal paths comparable to the expectation.
In this paper, we study new Cayley graphs over the additive group of Galois rings. First we prove that they are expander graphs by using a Weil-Carlitz-Uchiyama type estimation of character sums for Galois rings. We also show that Cayley graphs from Galois rings of characteristic 4 form a new infinite family of Ramanujan graphs by an elementary eigenvalue estimation. Moreover some other spectral properties of our graphs are also discussed.
Recently, several works by a number of authors have studied integrality, distance integrality, and distance powers of Cayley graphs over some finite groups, such as dicyclic groups and (generalized) dihedral groups. Our aim is to generalize and/or to give analogues of these results for generalized dicyclic groups. For example, we give a necessary and sufficient condition for a Cayley graph over a generalized dicyclic group to be integral (i.e., all eigenvalues of its adjacency matrix are in $\mathbb{Z}$). We also obtain sufficient conditions for the integrality of all distance powers of a Cayl…
We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $\frac{N}{N-1}\leq λ_N\leq 2$, to the case of chemical hypergraphs.
In this article, we establish some bounds involving the largest two distance Pareto eigenvalues of a connected graph. Also we characterize all possible values for smallest six distance Pareto eigenvalues of a connected graph.
We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.
The splitting field $\mathbb{SF}(Γ)$ of a mixed graph $Γ$ is the smallest field extension of $\mathbb{Q}$ which contains all eigenvalues of the Hermitian adjacency matrix of $Γ$. The extension degree $[\mathbb{SF}(Γ):\mathbb{Q}]$ is called the algebraic degree of $Γ$. In this paper, we determine the splitting fields and algebraic degrees of mixed Cayley graphs over abelian groups. This generalizes the main results of [K. Mönius, Splitting fields of spectra of circulant graphs, J. Algebra 594(15) (2022) 154--169] and [M. Kadyan, B. Bhattacharjya, Integral mixed Cayley graphs over abelian groups…
We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that $c_{2}\in [1/2, \sqrt{2+\sqrt{5}}]$, where $c_2$ is the minimum real number $c$ for which every real number greater than $c$ is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem.
