The first Zagreb index M 1 (G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M 2 (G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index M 1 (G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ), and minimum vertex degree (δ). Using this result, we find lower and upper bounds on M 2 (G). Also, we present lower and upper bounds on M 2 (G) + M 2 (G) in terms of n, m, Δ, and δ, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex M 1 (G) and second Zagreb coindex M 2 (G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.

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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