Lower Bounds for Inverse Sum Indeg Index of Graphs

I. Gutman, M. Matejić, E. Milovanović, I. Milovanović

Let $G=(V,E)$, $V=\{1,2,\ldots,n\}$, be a simple connected graph with $n$ vertices and $m$ edges and let $d_1\geq d_2\geq\cdots\geq d_n>0$, be the sequence of its vertex degrees. With $i\sim j$ we denote the adjacency of the vertices $i$ and $j$ in $G$. The inverse sum indeg index is defined as $ISI=\sum \frac{d_i\,d_j}{d_i+d_j}$ with summation going over all pairs of adjacent vertices. We consider lower bounds for $ISI$. We first analyze some lower bounds reported in the literature. Then we determine some new lower bounds.