Maximizing the Zagreb Indices of (n, m)-Graphs

For a (molecular) graph, the first and second Zagreb indices (M-1 and M-2) are two well-known topological indices in chemical graph theory introduced in 1972 by Gutman and Trinajstio. Let g(n,m) be the set of connected graphs of order n and with m edges. In this paper we characterize the extremal graphs from g(n,m) with n + 2 <= m <= 2n - 4 with maximal first Zagreb index and from g(n,m) with m - n = (k/2) - k for k >= 4 with maximal second Zagreb index, respectively. Finally a related conjecture has been proposed to the extremal graphs with respect to second Zagreb index.