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.

For a (molecular) graph \(G\) with vertex set \(V(G)\) and edge set \(E(G)\), the first and second Zagreb indices of \(G\) are defined as \(M_1(G) = \sum_{v \in V(G)} d_G(v)^2\) and \(M_2(G) = \sum_{uv \in E(G)} d_G(u)d_G(v)\), respectively, where \(d_G(v)\) is the degree of vertex \(v\) in \(G\). The alternative expression of \(M_1(G)\) is \(\sum_{uv \in E(G)}(d_G(u) + d_G(v))\). Recently Ashrafi, Došlić and Hamzeh introduced two related graphical invariants \(\overline{M_1}(G) = \sum_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{M_2}(G) = \sum_{uv \notin E(G)} d_G(u)d_G(v)\) named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants \(\overline{\Pi_1}(G) = \Pi_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{\Pi_2}(G) = \sum_{uv \notin E(G)} d_G(u)d_G(v)\) as the respective multiplicative versions of \(\overline{M_i}\) for \(i = 1, 2\). In this paper, we have reported some properties, especially upper and lower bounds, for these two graph invariants of connected (molecular) graphs. Moreover, some corresponding extremal graphs have been characterized with respect to these two indices