Padmakar–Ivan index of some families of graphs

Abstract

In this paper, we obtain the vertex Padmakar–Ivan index of [Formula: see text]-generalized join of graphs. As a consequence, we find the vertex Padmakar–Ivan index of the lexicographic product of graphs, the ideal-based zero-divisor graph and the zero-divisor graph of rings. In addition, we obtain the vertex Padmakar–Ivan index of the ideal-based zero-divisor graph and the zero-divisor graph of the ring of integers modulo [Formula: see text].