Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.
Classification :
05C50, 05C25
Mots-clés :
unitary Cayley graph, local ring, finite commutative ring, Ramanujan graph, energy of a graph, spectral moment
@article{10_37236_2390,
author = {Xiaogang Liu and Sanming Zhou},
title = {Spectral properties of unitary {Cayley} graphs of finite commutative rings},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {4},
doi = {10.37236/2390},
zbl = {1266.05082},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2390/}
}
TY - JOUR
AU - Xiaogang Liu
AU - Sanming Zhou
TI - Spectral properties of unitary Cayley graphs of finite commutative rings
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/2390/
DO - 10.37236/2390
ID - 10_37236_2390
ER -
%0 Journal Article
%A Xiaogang Liu
%A Sanming Zhou
%T Spectral properties of unitary Cayley graphs of finite commutative rings
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/2390/
%R 10.37236/2390
%F 10_37236_2390
Xiaogang Liu; Sanming Zhou. Spectral properties of unitary Cayley graphs of finite commutative rings. The electronic journal of combinatorics, Tome 19 (2012) no. 4. doi: 10.37236/2390
