The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

the sharp upper bounds and the sharp lower bounds of the largest eigenvalues $lambda_1$, the least eigenvalue $lambda_n$, the second largest eigenvalue $lambda_2$, the spread and the separator among all firefly graphs on $n$ vertices are determined.

let $n$ be any positive integer, the friendship graph $f_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. a graph $g$ is called cospectral with a graph $h$ if their adjacency matrices have the same eigenvalues. recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $g$ is any graph cospectral with $f_n$...

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

In this paper we investigate the least eigenvalue of a graph whose complement is connected, and present a lower bound for the least eigenvalue of such graph. We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.