On reduced second Zagreb index
The reduced second Zagreb index $RM_2$ R M 2 of a graph G is defined as $RM_2(G)\sum _{uv\in E(G)(d_G(u) 1)(d_G(v) 1) R M 2 G u v E G d G u 1 d G v 1 where $d_G(u) d G u is the degree of the vertex u of graph G Furtula et al. (Discrete Appl Math 178: 83 88, 2014) studied the difference between the classical Zagreb indices of graphs and showed that it is closely related to $RM_2$ R M 2 In this paper, we obtain an upper bound in terms of order n and size m on $RM_2$ R M 2 of $K_{r+1} K r 1 free graphs. Also we prove that among all graphs of order n with chromatic number \chi kh the Turan graph $T_{n,\chi T n kh is the unique graph having the maximum $RM_2$ R M 2 Furthermore, we completely characterize the extremal graphs with respect to $RM_2$ R M 2 among all unicyclic graphs of order n with girth g
J. Comb. Optim.