Geodetic spectra of graphs
Resource
European Journal of Combinatorics 25,383-391
Journal
European Journal of Combinatorics 25
Pages
383-391
Date Issued
2004
Date
2004
Author(s)
Chang, Gerard-J.
Tong, Li-Da
Wang, Hong-Tsu
DOI
246246/2006111501265209
Abstract
Geodetic numbers of graphs & digraphs have been investigated in the literature recently. The main purpose of this paper is to study the geodetic spectrum of a graph. For any two vertices u & v in an oriented graph D, a u–v geodesic is a shortest directed path from u to v. Let I (u, v) denote the set of all vertices lying on a u–v geodesic. For a vertex subset A, let I (A) denote the union of all I (u, v) for u, v ∈ A. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set A with I (A) = V(D). The (strong) geodetic spectrum of a graph G is the set of geodetic numbers of all (strongly connected) orientations of G. In this paper, we determine geodetic spectra & strong geodetic spectra of several classes of graphs. A conjecture & two problems given by Chartrand & Zhang are dealt with.
Subjects
Convex set
Geodesic
Geodetic number
Geodetic spectrum
Connected graph
Complete graph
Cycle
Tree
Complete r-partite graph
Type
journal article
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