On reduced second Zagreb index

Lkhagva Buyantogtokh (), Batmend Horoldagva () and Kinkar Chandra Das ()
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Lkhagva Buyantogtokh: Mongolian National University of Education
Batmend Horoldagva: Mongolian National University of Education
Kinkar Chandra Das: Sungkyunkwan University

Journal of Combinatorial Optimization, 2020, vol. 39, issue 3, No 8, 776-791

Abstract: Abstract The reduced second Zagreb index $$RM_2$$RM2 of a graph G is defined as $$RM_2(G)=\sum _{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)$$RM2(G)=∑uv∈E(G)(dG(u)-1)(dG(v)-1), where $$d_G(u)$$dG(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 $$RM_2$$RM2. In this paper, we obtain an upper bound in terms of order n and size m on $$RM_2$$RM2 of $$K_{r+1}$$Kr+1-free graphs. Also we prove that among all graphs of order n with chromatic number $$\chi $$χ, the Turán graph $$T_{n,\,\chi }$$Tn,χ is the unique graph having the maximum $$RM_2$$RM2. Furthermore, we completely characterize the extremal graphs with respect to $$RM_2$$RM2 among all unicyclic graphs of order n with girth g.

Keywords: Vertex degree; Zagreb indices; Chromatic number; Turán graph; Girth; 05C15; 05C35; 05C40; 05C69 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s10878-019-00518-7

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