Sharp bounds on the reduced second Zagreb index of graphs with given number of cut vertices

The first Zagreb index M-1 of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index M-2 of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices. In Furtula et al. (2014) Furtula, Gutman and Ediz first proposed the reduced second Zagreb index as RM2(G) = Sigma(uv is an element of E(G)) (d(u) - 1)(d(v) - 1) = Delta M(G) + m(G), where Delta M(G) is the difference between M-2 and M-1 of graph G and m(G) is the size of G. In this paper, some graph transformations are introduced. We study the monotonicity of RM2(G) with respect to these graph transformations. Then based on these properties, we present sharp upper and lower bounds on RM2(G) of connected n-vertex cyclic graphs with given number of cut vertices. All the corresponding extremal graphs are identified. (C) 2019 Elsevier B.V. All rights reserved.