On Inverse degree and topological indices of graphs

Abstract

Let $G=(V,E)$ be a simple graph of order $n$ and size $m$ with maximum degree $\Delta$ and minimum degree $\delta$\,. The inverse degree of a graph $G$ with no isolated vertices is defined as $$ID(G)=\sum\limits^n_{i=1}\frac{1}{d_i}\,,$$
where $d_i$ is the degree of the vertex $v_i\in V(G)$\,. In this paper, we obtain several lower and upper bounds on $ID(G)$ of graph $G$ and characterize graphs for which these bounds are best possible. Moreover, we compare between inverse degree $ID(G)$ and topological indices ($GA_1$-index, $ABC$-index, $Kf$-index) of graphs.