Abstract
LetG be a simple connected graph. Theeccentricity e(u) of a vertexu inG is given bye(u)=max {d(u,v): vεV(G)}. A vertexv is called aperipheral vertex ofG ife(v)=dia(G). A vertexv is called aneccentric vertex ofG ifd(v,c)=e(C) for some center vertexc ofG. LetP(G) andEC(G) denote the sets of peripheral vertices and eccentric vertices ofG, respectively. The main result in this paper is a discription of general classes of graphs for whichP(G)=EC(G). In particular, we give necessary and sufficient conditions for a graphG withdia(G)=2r(G) ordia(G)=2r(G)−1 to satisfyP(G)=EC(G). Also, we present several graphs which then are used to show that all possible set-inclusion relations betweenP(G) andEC(G) may occur.
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Research supported in part by Contract 86-LBR(39)-039-05 from the Louisiana Education Quality Support Fund.
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Reid, K.B., Weizhen, G. Peripheral and eccentric vertices in graphs. Graphs and Combinatorics 8, 361–375 (1992). https://doi.org/10.1007/BF02351592
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DOI: https://doi.org/10.1007/BF02351592