ECCENTRIC SPECTRUM OF A GRAPH

Abstract

In the related literatures, the eccentricities of graphs have been studied recently. The main purpose of this paper is to discuss the eccentric spectrum of a graph. For any two vertices $u$ and $v$ in a connected graph $G$, $d_G(u,v)$ denotes the distance between vertices $u$ and $v$. The eccentricity $e_G(v)$ of a vertex $v$ in $G$ is the maximum number of $d_G(v,u)$ over all vertex $u$. A vertex $u$ is an eccentric vertex if there exists a vertex $v$ such that $e_G(v)=d_G(v,u)$. A number $k$ is called an eccentric number of $G$ if, for each vertex $v$ with $e_G(v)=k$, $v$ is an eccentric vertex. The eccentric spectrum $S_G$ of a connected graph $G$ is a set of all eccentric numbers in $G$. If $d$ is the diameter of $G$, then $d\in S_G$. In the paper, we show that for positive integers $r\leq d\leq 2r$ and $d\in S\subseteq \{r,r+1,...,d\}$, there exists a connected graph $G$ with radius $r$, diameter $d$ and eccentric spectrum $S$. This result also proves the conjecture of Chartrand, Gu, Schultz, and Winters in [4].