THE ZAGREB INDICES OF BIPARTITE GRAPHS WITH MORE EDGES

Abstract

For a (molecular) graph, the first and second Zagreb indices (M₁ and M₂) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajsti&#x0107;. The first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let <TEX>$K_{n_1,n_2}^{P}$</TEX> with n₁ <TEX>$\leq$</TEX> n₂, n₁ + n₂ = n and p < n₁ be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph K_n1,n2. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from <TEX>$K_{n_1,n_2}^{P}$</TEX> 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 <TEX>$K_{n_1,n_2}^{P}$</TEX> with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.