Bicyclic graphs with maximal revised Szeged index

Abstract

The revised Szeged index $Sz^*(G)$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ 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 $n_0(e)$ is the number of vertices equidistant to $u$ and $v$. Hansen used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph $G$ of order $n \geq 6$: $$ Sz^*(G)\leq \{{array}{ll} (n^3+n^2-n-1)/4,& {if $n$ is odd}, (n^3+n^2-n)/4, & {if $n$ is even}. {array}. $$ with equality if and only if $G$ is the graph obtained from the cycle $C_{n-1}$ by duplicating a single vertex. This paper is to give a confirmative proof to this conjecture.