Sharp upper bounds for Zagreb indices of bipartite graphs with a given diameter

For a (molecular) graph, the first Zagreb index M1M1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M2M2 is equal to the sum of products of the degrees of a pair of adjacent vertices. In this work, we study the Zagreb indices of bipartite graphs of order nn with diameter dd and sharp upper bounds are obtained for M1(G)M1(G) and M2(G)M2(G) with G∈ℬ(n,d)G∈ℬ(n,d), where ℬ(n,d)ℬ(n,d) is the set of all the nn-vertex bipartite graphs with diameter dd. Furthermore, we study the relationship between the maximal Zagreb indices of graphs in ℬ(n,d)ℬ(n,d) and the diameter dd. As a consequence, bipartite graphs with the largest, second-largest and smallest Zagreb indices are characterized.