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Title: Graphs with the same peripheral and center eccentric vertices (English)
Author: Kyš, Peter
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 125
Issue: 3
Year: 2000
Pages: 331-339
Summary lang: English
Category: math
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 $\mathop PeriG)$. 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. (English)
Keyword: graph
Keyword: radius
Keyword: diameter
Keyword: center
Keyword: eccentricity
Keyword: distance
MSC: 05C12
MSC: 05C20
MSC: 05C75
idZBL: Zbl 0963.05046
idMR: MR1790124
DOI: 10.21136/MB.2000.126124
Date available: 2009-09-24T21:44:02Z
Last updated: 2020-07-29
Stable URL:
Reference: [1] Chаrtrаnd G., Lesniаk L.: Graphs and Digraphs.Wadsworth and Brooks, Monterey, California, 1986.
Reference: [2] Buckley F., Lewïnter M.: Minimal graph embeddings, eccentric vertices and the peripherian.Proc. Fifth Carribean Conference on Cornbinatorics and Computing. University of the West Indies, 1988, pp. 72-84.
Reference: [3] Buckley P., Lewinter M.: Graphs with all diametral paths through distant central vertices.Math. Comput. Modelling 17 (1993), 35-41. MR 1236507, 10.1016/0895-7177(93)90250-3
Reference: [4] Gliviаk F.: Two classes of graphs related to extrernal eccentricities.Math. Bohem. 122 (1997), 231-241. MR 1600875
Reference: [5] Ore O.: Diameters in graphs.J.Combin.Theory 5 (1968), 75-81. Zbl 0175.20804, MR 0227043, 10.1016/S0021-9800(68)80030-4


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