General degree-eccentricity index of unicyclic graphs of given order, girth and maximum degree

Abstract

For a connected graph $G$ and $a,b\in R$, the general degree-eccentricity index of $G$ is defined as $DE{I}_{a,b}\left(G\right)={\sum }_{v\in V\left(G\right)}{d}_G^a\left(v\right)ec{c}_G^b\left(v\right)$, where $V\left(G\right)$ is the vertex set of $G$, $d_G\left(v\right)$ is the degree of a vertex $v$ and $ec{c}_G\left(v\right)$ is the eccentricity of $v$ in $G$, i.e. the maximum distance from $v$ to another vertex of the graph. This index generalizes several well-known ‘topological indices’ of graphs such as the eccentric connectivity index. We characterize the unique unicyclic graphs with the maximum and the minimum general degree-eccentricity index among all $n$-vertex unicyclic graphs with fixed order, girth, and maximum degree for the cases $a\ge 1,b\le 0$ and $0\le a\le 1,b\ge 0$.