On General Reduced Second Zagreb Index of Graphs

Lkhagva Buyantogtokh, Batmend Horoldagva and Kinkar Chandra Das ()
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Lkhagva Buyantogtokh: Department of Mathematics, Mongolian National University of Education, Baga Toiruu-14, Ulaanbaatar 210648, Mongolia
Batmend Horoldagva: Department of Mathematics, Mongolian National University of Education, Baga Toiruu-14, Ulaanbaatar 210648, Mongolia
Kinkar Chandra Das: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea

Mathematics, 2022, vol. 10, issue 19, 1-18

Abstract: Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant G R M α , known under the name general reduced second Zagreb index, is defined as G R M α ( Γ ) = ∑ u v ∈ E ( Γ ) ( d Γ ( u ) + α ) ( d Γ ( v ) + α ) , where d Γ ( v ) is the degree of the vertex v of the graph Γ and α is any real number. In this paper, among all trees of order n , and all unicyclic graphs of order n with girth g , we characterize the extremal graphs with respect to G R M α ( α ≥ − 1 2 ) . Using the extremal unicyclic graphs, we obtain a lower bound on G R M α ( Γ ) of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on G R M α of different classes of graphs in terms of order n , size m , independence number γ , chromatic number k , etc. In particular, we present an upper bound on G R M α of connected triangle-free graph of order n > 2 , m > 0 edges with α > − 1.5 , and characterize the extremal graphs. Finally, we prove that the Turán graph T n ( k ) gives the maximum G R M α ( α ≥ − 1 ) among all graphs of order n with chromatic number k .

Keywords: Zagreb indices; girth; clique number; chromatic number; Turán graph (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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