Bicyclic graphs with maximal revised Szeged index

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Abstract

The revised Szeged index of a graph G is defined as Sz(G)=e=uvE(nu(e)+n0(e)/2)(nv(e)+n0(e)/2), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u, and n0(e) is the number of vertices equidistant to u and v. Hansen et al. used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph G of order n6: Sz(G){(n3+n2n1)/4,if  n  is odd ,(n3+n2n)/4,if  n  is even . with equality if and only if G is the graph obtained from the cycle Cn1 by duplicating a single vertex. This paper is to give a confirmative proof to this conjecture.

Keywords

Wiener index
Szeged index
Revised Szeged index
Bicyclic graph

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Supported by NSFC and the “973” program.