General Zagreb indices of composite graphs

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