The Zagreb Indices of Bipartite Graphs With More Edges

Abstract

For a (molecular) graph, the first and second Zagreb indices (M 1 and M 2 ) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M 1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M 2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let K p n 1 ,n 2 with n 1 ≤ n 2 , n 1 + n 2 = n and p < n 1 be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph Kn 1 ,n 2 . In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from K p n 1 ,n 2 and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from K p n 1 ,n 2 with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.

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