General Zagreb indices of composite graphs

Abstract

The first and second general Zagreb indices, M? 1 and M? 2, are the sum of the terms ? (u) ?+ ? (v) ? and ? (u) ?� ? (v) ?, respectively, over all pairs of adjacent vertices u, v of a graph, where ? (x) is the degree of the vertex x, and ? is a real number. For ?= 1, M? 1 and M? 2 are equal to the ordinary first and second Zagreb indices. For some other values of ?, M? 1 and M? 2 reduce to a variety of other, earlier considered, topological indices. In this paper, we establish expressions for M? 1 and M? 2 for several types of composite graphs, and give examples pointing at possible applications of these expressions.