Eccentric spectrum of a graph (Q947048)

In the related literature, 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 \(d\in S\subseteq \{r,r+1,\dots,d\}\), there exists a connected graph \(G\) with radius \(r\), diameter \(d\) and eccentric spectrum \(S\). This result also proves the conjecture of \textit{G. Chartrand, M. Schultz} and \textit{S. J. Winters} in [Networks 28, No.4, 181-186 (1996; Zbl 0873.90099)].