On Zagreb indices of graphs

The first Zagreb index M1(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M2(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 M1(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 M2(G). Also, we present lower and upper bounds on $$M_2 (G) + M_2 (\bar G)$$ in terms of n, m, Δ, and δ, where $$\bar G$$ denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex $$\bar M_1 (G)$$ and second Zagreb coindex $$\bar M_2 (G)$$. Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.

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