Harmonic and Geometric-Arithmetic Indices of Boolean Function Graph $B(K_p, INC,\overline{K_q})$

Abstract

For any graph G, let $V(G)$ and $E(G)$ denote the vertex set and edge set of G respectively. The Harmonic index $H(G)$ of a graph G is defined as the sum of the weights $\frac{2}{d(u)+d(v)} $ of all edges uv of G and the Geometric-Arithmetic index $GA(G)$ of G is defined as the sum of the weights $\frac{2\sqrt{d(u)d(v)} }{d(u)+d(v)} $, where $d(u)$ denotes the degree of a vertex u in G. The Boolean function graph $B(K_p, INC,\overline{K_ q})$ of G is a graph with vertex set $V(G)\cup E(G)$ and two vertices in $B(K_p, INC,\overline{K_ q})$ are adjacent if and only if they correspond to two adjacent vertices of G, two nonadjacent vertices of G or to a vertex and an edge incident to it in G, For brevity, this graph is denoted by $B_{4}(G)$. In this paper, lower and upper bounds of $H(B_{4}(G))$ and $GA(B_{4}(G))$ are obtained. These indices are found for some particular graphs.