On acyclic and unicyclic conjugated graphs with maximum Zagreb indices

Abstract

For a molecular graph, the first Zagreb index M₁ is equal to the sum of squares of the vertex degrees, and the second Zagreb index M₂ is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we first obtain a sharp upper bound on M₁-values of trees in terms of the order and the given size of matching. Then we investigate Zagreb indices of unicyclic conjugated graphs G (i.e., unicyclic graphs with a perfect matching) and a sharp upper bound on M₁(G), M₂ (G) are determined, respectively. The sharp upper bounds for Zagreb indices of unicyclic graphs in terms of the order and given size of matching are also determined.