Spectral properties of inverse sum indeg index of graphs
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Spectral properties of inverse sum indeg index of graphs Fengwei Li1,3 · Xueliang Li2 · Hajo Broersma3 Received: 8 January 2020 / Accepted: 4 September 2020 © Springer Nature Switzerland AG 2020
Abstract The inverse sum indeg (ISI) index is a vertex-degree-based topological index that was selected by Vukičević and Gašperov in 2010 as a significant predictor of the total surface area of octane isomers. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the ISI index from an algebraic viewpoint. We introduce suitably modified versions of the classical adjacency matrix and the Laplacian matrix involving the degrees of the vertices of a graph. Moreover, we formulate the ISI index in terms of these matrices. Keywords Inverse sum indeg index · Algebraic properties · Laplacian matrix · Laplacian eigenvalues · Topological index Mathematics Subject Classification 05C35 · 05C90
1 Introduction Topological indices, which can capture some of the properties of a molecule (or a graph) in a single number, have a prominent place in theoretical chemistry, pharmacology and biology, etc.; see [7, 39] for examples. Over the years, several dozens of topological indices have been proposed and studied in these fields.
* Fengwei Li [email protected] Xueliang Li [email protected] Hajo Broersma [email protected] 1
College of Basic Science, Ningbo University of Finance & Economics, Ningbo 315175, Zhejiang, China
2
Center for Combinatorics, Nankai University, Tianjin 300071, China
3
Faculty of EEMCS, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands
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Vol.:(0123456789)
Journal of Mathematical Chemistry
Topological indices depending on end-vertex degrees of edges are called vertex-degree-based topological indices (VDB topological indices for short) [16, 31]. Probably, the Randić index is the best known VDB topological index [15, 18, 30, 32]. It is named after its inventor Milan Randić [32] who defined it in 1976 as
R(G) =
�
1 , √ d uv∈E(G) u dv
where du and dv denote the degree of the end-vertices u and v of the edge uv ∈ E(G) for a graph G. During many years, scientists have been trying to improve the predictive power of the Randić index. This has led to the introduction of numerous new topological indices, resembling the original Randić index [3, 4, 6, 9, 21–27, 38, 41]. Let G be an undirected, finite, and simple connected graph with vertex set V(G) and edge set E(G). For u ∈ V(G) , N(u) denotes the set of neighbors of u in G, and the degree of u is du = |N(u)|. The Zagreb indices are among the oldest topological indices, and were introduced by Gutman and Trinajstić [17] in 1972. These indices have since been used to study molecular complexity, chirality, and hetero-systems. The first and second Zagreb indices of G are denoted by M1 (G) and M2 (G) , respectively, and defined as ∑ ∑ M1 (G) = du2 , M2 (G) = du dv . u∈V(G)
uv∈E(G)
A closel
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