## 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

### Keywords:

graph; radius; diameter; center; eccentricity; distance
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