Title:

Graphs with the same peripheral and center eccentric vertices (English) 
Author:

Kyš, Peter 
Language:

English 
Journal:

Mathematica Bohemica 
ISSN:

08627959 (print) 
ISSN:

24647136 (online) 
Volume:

125 
Issue:

3 
Year:

2000 
Pages:

331339 
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 Sgraph if $\mathop Cep(G) = \mathop Peri(G)$. In this paper we characterize Sgraphs with diameters less or equal to four, give some constructions of Sgraphs and investigate Sgraphs 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:

20090924T21:44:02Z 
Last updated:

20200729 
Stable URL:

http://hdl.handle.net/10338.dmlcz/126124 
. 
Reference:

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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. 7284. 
Reference:

[3] Buckley P., Lewinter M.: Graphs with all diametral paths through distant central vertices.Math. Comput. Modelling 17 (1993), 3541. MR 1236507, 10.1016/08957177(93)902503 
Reference:

[4] Gliviаk F.: Two classes of graphs related to extrernal eccentricities.Math. Bohem. 122 (1997), 231241. MR 1600875 
Reference:

[5] Ore O.: Diameters in graphs.J.Combin.Theory 5 (1968), 7581. Zbl 0175.20804, MR 0227043, 10.1016/S00219800(68)800304 
. 