The second Zagreb index of acyclic conjugated molecules

The second Zagreb index M2(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv in G, where d(u) denotes the degree of the vertex u. In this paper, we give a sharp upper bound on the second Zagreb index of conjugated trees (trees with a perfect matching) in terms of the number of vertices. A sharp upper bound on the second Zagreb index of trees with a given size of matching is also given.