Elsevier

Discrete Applied Mathematics

Volume 158, Issue 3, 6 February 2010, Pages 219-221
Discrete Applied Mathematics

Note
A characterization of block graphs

https://doi.org/10.1016/j.dam.2009.09.024Get rights and content
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Abstract

A block graph is a graph whose blocks are cliques. For each edge e=uv of a graph G, let Ne(u) denote the set of all vertices in G which are closer to u than v. In this paper we prove that a graph G is a block graph if and only if it satisfies two conditions: (a) The shortest path between any two vertices of G is unique; and (b) For each edge e=uvE(G), if xNe(u) and yNe(v), then, and only then, the shortest path between x and y contains the edge e. This confirms a conjecture of Dobrynin and Gutman [A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math., Beograd. 56 (1994) 18–22].

Keywords

Block graphs
2-connected graphs
Graph invariants
Wiener index

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