Study Some of Algebraic Properties of a Class of Circulant Graphs

Let $Lambda=Cay(mathbb{Z}_{n}, Omega)$ be the Cayley graph on the cyclic group $mathbb{Z}_{n}$, where $n=2m$, $m$ is an odd integer, and $Omega={{t}inmathbb{Z}_{n},|,, ext {t is odd and $teq m$}}$ is an inverse closed subset of $mathbb{Z}_{n}-{0}$. In this paper, we show that $Aut(Lambda)congmathbb{Z}_2 imes Sym(m)$. Also, it is shown that $Lambda$ is an integral graph, in fact the adjacency spectrum of $Lambda$ is $m-1, 1-m, 1^{(m-1)}, -1^{(m-1)}$, where the superscripts give the multiplicities of eigenvalues with multiplicity greater than one. Finally, we prove that multicone graph $K_v igtriangledownLambda$ is determined by its adjacency spectrum, where $K_v$ is the complete graph on $v$ vertices.