Journal of Combinatorial Optimization https://doi.org/10.1007/s10878-019-00518-7 On reduced second Zagreb index Lkhagva Buyantogtokh 1 · Batmend Horoldagva 1,2 · Kinkar Chandra Das 3 © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract The reduced second Zagreb index RM 2 of a graph G is defined as RM 2 (G) = ∑ uv∈E (G) (d G (u ) − 1)(d G (v) − 1), where 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 . In this paper, we obtain an upper bound in terms of order n and size m on RM 2 of K r +1 -free graphs. Also we prove that among all graphs of order n with chromatic number χ , the Turán graph T n,χ is the unique graph having the maximum RM 2 . Furthermore, we completely characterize the extremal graphs with respect to RM 2 among all unicyclic graphs of order n with girth g. Keywords Vertex degree · Zagreb indices · Chromatic number · Turán graph · Girth Mathematics Subject Classification 05C15 · 05C35 · 05C40 · 05C69 1 Introduction Let G = (V , E ) be a graph with vertex set V (G) and edge set E (G). Denote by d G (u ), the degree of the vertex u of G. A pendant vertex is a vertex of degree one. An edge of a graph is said to be pendant if one of its end vertices is pendant. For v ∈ V (G), B Kinkar Chandra Das kinkardas2003@gmail.com Lkhagva Buyantogtokh buyantogtokh.l@msue.edu.mn Batmend Horoldagva horoldagva@msue.edu.mn 1 Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia 2 Institute of Mathematics, National University of Mongolia, Baga toiruu-1, Ulaanbaatar, Mongolia 3 Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea 123