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

For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M 2 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 n with diameter d and sharp upper bounds are obtained for M 1 ( G ) and M 2 ( G ) with G ∈ ℬ ( n , d ) , where ℬ ( n , d ) is the set of all the n -vertex bipartite graphs with diameter d . Furthermore, we study the relationship between the maximal Zagreb indices of graphs in ℬ ( n , d ) and the diameter d . As a consequence, bipartite graphs with the largest, second-largest and smallest Zagreb indices are characterized.