Zagreb indices of graphs

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 (Delta), and minimum vertex degree (delta). 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) over bar) in terms of n, m, Delta, and delta, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex (M) over bar (1)(G) and second Zagreb coindex (M) over bar (2)(G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.