On general reduced second Zagreb index of graphs

Recently, Furtula et al. [B. Furtula, I. Gutman, S. Ediz, On difference of Zagreb indices, Discrete Appl. Math., 2014] introduced a new vertex-degree-based graph invariant "reduced second Zagreb index" in chemical graph theory. Here we generalize the reduced second Zagreb index (call "general reduced second Zagreb index"), denoted by GRM_α(G) and is defined as: GRM_α(G) = ^∑ _uv∈E(G₎(d_G(u) + α)(d_G(v) + α), where α is any real number and d_G(v) is the degree of the vertex v of G. Let G^k _n be the set of connected graphs of order n with k cut edges. In this paper, we study some properties of GRM_α(G) for connected graphs G. Moreover, we obtain the sharp upper bounds on GRM_α(G) in G^k _n for α ≥ −1/2 and characterize the extremal graphs.