ﻻ يوجد ملخص باللغة العربية
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 unitary 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.
Following a problem posed by Lovasz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that is, for grou
A graph $G$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {em bi-Cayley graph/} over $H$. Such a graph $G$ is called {em normal/} if $H$ is normal in the full automorphism group of $G
In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In
In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.
Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X subseteq mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ b