On Bounds for Harmonic Topological Index

Abstract

Let $G= (V,E)$, $V=\{1,2,\ldots , n\}$, $E=\{e_1,e_2,\ldots , e_m\}$, be a simple graph with $n$ vertices and $m$ edges.  Denote by $d_1\ge d_2 \ge \cdots \ge d_n>0$  and $d(e_1) \ge d(e_2) \ge \cdots \ge d(e_m)$,  sequences of  vertex and edge degrees, respectively. If $i$-th and $j$-th vertices of the graph $G$ are adjacent,  it is denoted as $i\sim j$. Graph invariants referred to as harmonic index is defined as $H(G) = \displaystyle\sum_{i\sim j}\frac{2}{d_i+d_j}$. Lower and upper  bounds for invariants  $H(G)$ are obtained.