The purpose of this paper is to study spectral properties of a family of Cayley graphs on finite commutative rings. Let $R$ be such a ring and $R^times$ its set of units. Let $Q_R={u^2: uin R^times}$ and $T_R=Q_Rcup(-Q_R)$. We define the quadratic un
itary Cayley graph of $R$, denoted by $mathcal{G}_R$, to be the Cayley graph on the additive group of $R$ with respect to $T_R$; that is, $mathcal{G}_R$ has vertex set $R$ such that $x, y in R$ are adjacent if and only if $x-yin T_R$. It is well known that any finite commutative ring $R$ can be decomposed as $R=R_1times R_2timescdotstimes R_s$, where each $R_i$ is a local ring with maximal ideal $M_i$. Let $R_0$ be a local ring with maximal ideal $M_0$ such that $|R_0|/|M_0| equiv 3,(mod,4)$. We determine the spectra of $mathcal{G}_R$ and $mathcal{G}_{R_0times R}$ under the condition that $|R_i|/|M_i|equiv 1,(mod,4)$ for $1 le i le s$. We compute the energies and spectral moments of such quadratic unitary Cayley graphs, and determine when such a graph is hyperenergetic or Ramanujan.
Francis Castro, et al [2] computed the exact divisibility of families of exponential sums associated to binomials $F(X) = aX^{d_1} + bX^{d_2}$ over $mathbb{F}_p$, and a conjecture is presented for related work. Here we study this question.
Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices respectively. The dumbbell graph, denoted by $D_{p,k,q}$, is the graph obtained from two cycles $C_p$, $C_q$ and a path $P_{k+2}$ by identifying each pendant vertex of $P_{k+2}$ with a ver
tex of a cycle respectively. The theta graph, denoted by $Theta_{r,s,t}$, is the graph formed by joining two given vertices via three disjoint paths $P_{r}$, $P_{s}$ and $P_{t}$ respectively. In this paper, we prove that all dumbbell graphs as well as theta graphs are determined by their Laplacian spectra.
The subdivision graph $mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1dot{v
ee}G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$. The emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by $G_1underline{vee}G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, where $I(G_1)$ is the set of inserted vertices of $mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1dot{vee}G_2$ (respectively, $G_1underline{vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$, in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, these results enable us to construct infinitely many pairs of cospectral graphs. We also give the number of the spanning trees and the Kirchhoff index of $G_1dot{vee}G_2$ (respectively, $G_1underline{vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$.
