Peripheral and eccentric vertices in graphs
type: Item
instance of: scholarly article
title: Peripheral and eccentric vertices in graphs
zbMATH Open document ID: 0777.05056
DOI: 10.1007/BF02351592
author: Weizhen Gu
author: K. B. Reid
published in: Graphs and Combinatorics
publication date: 1993-04-01T00:00:00Z
review text: Let \(r\) and \(d\) denote the radius and diameter of a graph \(G\) and \(\text{P}(G)\) denote the set of peripheral vertices of \(G\). A vertex \(v\) such that \(d(v,c)=r\) for some central vertex \(c\) of \(G\), is defined, by the authors, to be an eccentric vertex. Let \(\text{EC}(G)\) denote the set of eccentric vertices of \(G\). Using some interesting constructions the authors prove, in the last section, that all types of set inclusion relations are possible between \(\text{P}(G)\) and \(\text{EC}(G)\). In section 1 they prove that \(\text{EC}(G)=\text{P}(G)\) when either (i) \(G\) is a simple connected graph with a unique central vertex \(v\), such that every block containing \(v\) is complete or (ii) \(G\) is a non-selfcentered graph with \(\text{C}(G)\) contained in a complete block, and \(|\text{C}(G)|\geq 2\). In the middle section they obtain necessary and sufficient conditions for \(\text{EC}(G)=\text{P}(G)\) when \(d=2r\) or \(d=2r-1\).
Mathematics Subject Classification ID: 05C12
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 05C35
zbMATH DE Number: 147136
zbMATH Keywords: radius
zbMATH Keywords: diameter
zbMATH Keywords: peripheral vertices
zbMATH Keywords: eccentric vertex
MaRDI profile type: Publication
cites work: Diameters in graphs
cites work: Unique eccentric point graphs
full work available at URL: https://doi.org/10.1007/bf02351592
OpenAlex ID: W1969794462
description: scientific article; zbMATH DE number 147136