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Graphs with the same peripheral and center eccentric vertices. (English) Zbl 0963.05046

Summary: The eccentricity \(e(v)\) of a vertex \(v\) is the distance from \(v\) to a vertex farthest from \(v\), and \(u\) is an eccentric vertex for \(v\) if its distance from \(v\) is \(d(u,v) = e(v)\). A vertex of maximum eccentricity in a graph \(G\) is called peripheral, and the set of all such vertices is the peripherian, denoted \(\text{Peri}(G)\). We use \(\text{Cep}(G)\) to denote the set of eccentric vertices of central vertices in \(C(G)\). A graph \(G\) is called an S-graph if \(\text{Cep}(G)=\text{Peri}(G)\). In this paper we characterize S-graphs with diameters less or equal to four, give some constructions of S-graphs and investigate S-graphs with one central vertex. We also correct and generalize some results of F. Gliviak.

MSC:

05C12 Distance in graphs
05C75 Structural characterization of families of graphs
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