Some properties of the inverse degree index and coindex of trees

Abstract

Let $G=(V,E)$, $V=\{v_1,v_2,\ldots , v_n\}$, be a simple graph without isolated vertices, with the sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n>0$, $d_i=d(v_i)$,. If vertices $v_i$ and $v_j$ are adjacent in $G$, we write $i\sim j$, otherwise we write $i\nsim j$. The inverse degree topological index of $G$ is defined to be $ID(G)=\sum_{i=1}^n \frac{1}{d_i}= \sum_{i\sim j}\left(\frac{1}{d_i^2}+ \frac{1}{d_j^2}\right)$, and the inverse degree coindex  $\overline{ID}(G)=\sum_{i\nsim j} \left(\frac{1}{d_i^2}+ \frac{1}{d_j^2}\right)$. We obtain a number of inequalities which determine bounds for the $ID(G)$ and $\overline{ID}(G)$ when $G$ is a tree.