Graph Theory With Applications
Graph Theory With Applications
The connected p-center problem on block graphs with forbidden vertices
Theoretical Computer Science
Improved bounds on the difference between the Szeged index and the Wiener index of graphs
European Journal of Combinatorics
| Hi-index | 0.05 |
A block graph is a graph whose blocks are cliques. For each edge e=uv of a graph G, let N"e(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=uv@?E(G), if x@?N"e(u) and y@?N"e(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].