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MATHEMATICA BOHEMICA, Vol. 125, No. 3, pp. 331-339 (2000)
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# Graphs with the same peripheral and

center eccentric vertices

## Peter Kys

* Peter Kys*, Faculty of Mathematics and Physics, Mlynska dolina, 842 15 Bratislava, Slovakia, e-mail: ` kys@fmph.uniba.sk`

**Abstract:**
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 $\mathop PeriG)$. We use $\mathop Cep(G)$ to denote the set of eccentric vertices of vertices in $C(G)$. A graph $G$ is called an S-graph if $\mathop Cep(G) = \mathop Peri(G)$. 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.

**Keywords:** graph, radius, diameter, center

**Classification (MSC2000):** 05C12, 05C20

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