Vertex PI Indices of Some Sums of Graphs

Abstract

The vertex Padmakar-Ivan \((PI_v)\) index of a graph \(G\) is defined as the summation of the sums of \([m_{eu}(e|G) + m_{eu}(e|G)]\) over all edges \(e = uv\) of a connected graph \(G\), where \(m_{eu}(e|G)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\), and \(m_{eu}(e|G)\) is the number of vertices of \(G\) lying closer to \(v\) than to \(u\). In this paper, we give the explicit expressions of the vertex PI indices of some sums of graphs.