A relation between a vertex-degree-based topological index and its energy

Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

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Relating the ABC and harmonic indices

Journal of the Serbian Chemical Society ◽

10.2298/jsc130930001g ◽

2014 ◽

Vol 79 (5) ◽

pp. 557-563 ◽

Cited By ~ 4

Author(s):

Ivan Gutman ◽

Lingping Zhong ◽

Kexiang Xu

Keyword(s):

Molecular Structure ◽

Topological Index ◽

Molecular Graph ◽

Vertex Degree ◽

Harmonic Index ◽

Abc Index ◽

Atom Bond ◽

Structure Descriptor

The atom-bond connectivity (ABC) index is a much-studied molecular structure descriptor, based on the degrees of the vertices of the molecular graph. Recently, another vertex-degree-based topological index - the harmonic index (H) - attracted attention and gained popularity. We show how ABC and H are related.

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ORDERING CATACONDENSED HEXAGONAL SYSTEMS WITH RESPECT TO VDB TOPOLOGICAL INDICES

Revista de Matemática Teoría y Aplicaciones ◽

10.15517/rmta.v23i1.22554 ◽

2017 ◽

Vol 23 (1) ◽

pp. 277-289

Author(s):

Juan Rada

Keyword(s):

Topological Index ◽

Topological Indices ◽

Vertex Degree ◽

Extremal Values ◽

Hexagonal Systems

In this paper we give a complete description of the ordering relations in the set of catacondensed hexagonal systems, with respect to a vertex-degree-based topological index. As a consequence, extremal values of vertex-degree-based topological indices in special subsets of the set of catacondensed hexagonal systems are computed.

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Extended energy and its dependence on molecular structure

Canadian Journal of Chemistry ◽

10.1139/cjc-2016-0636 ◽

2017 ◽

Vol 95 (5) ◽

pp. 526-529 ◽

Cited By ~ 8

Author(s):

Ivan Gutman ◽

Boris Furtula ◽

Kinkar Ch. Das

Keyword(s):

Molecular Structure ◽

Electron Energy ◽

Topological Index ◽

Molecular Properties ◽

Vertex Degree ◽

Energy Formula ◽

The Difference ◽

Basic Characteristics ◽

Ga Index ◽

Structure Descriptor

The extended energy ([Formula: see text]) is a vertex degree based and spectrum-based molecular structure descriptor, shown to be well correlated with a variety of physicochemical molecular properties. We investigate the dependence of [Formula: see text] on molecular structure and establish its basic characteristics. In particular, we show how [Formula: see text] is related with the geometric–arithmetic (GA) topological index. Our main finding is that the difference between [Formula: see text] and the total π-electron energy is linearly proportional to the difference between the number of edges and the GA index.

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Spectrum and energy of the Sombor matrix

Vojnotehnicki glasnik ◽

10.5937/vojtehg69-31995 ◽

2021 ◽

Vol 69 (3) ◽

pp. 551-561

Author(s):

Ivan Gutman

Keyword(s):

Spectral Theory ◽

Spectral Properties ◽

Topological Index ◽

Vertex Degree ◽

Graph Invariants

Introduction/purpose: The Sombor matrix is a vertex-degree-based matrix associated with the Sombor index. The paper is concerned with the spectral properties of the Sombor matrix. Results: Equalities and inequalities for the eigenvalues of the Sombor matrix are obtained, from which two fundamental bounds for the Sombor energy (= energy of the Sombor matrix) are established. These bounds depend on the Sombor index and on the "forgotten" topological index. Conclusion: The results of the paper contribute to the spectral theory of the Sombor matrix, as well as to the general spectral theory of matrices associated with vertex-degree-based graph invariants.

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Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

Symmetry ◽

10.3390/sym13101903 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1903

Author(s):

Juan Monsalve ◽

Juan Rada

Keyword(s):

Lower Bounds ◽

Topological Index ◽

General Setting ◽

Upper Bounds ◽

Topological Indices ◽

Vertex Degree ◽

Upper And Lower Bounds ◽

Lower And Upper Bounds ◽

Extremal Values

A vertex-degree-based (VDB, for short) topological index φ induced by the numbers φij was recently defined for a digraph D, as φD=12∑uvφdu+dv−, where du+ denotes the out-degree of the vertex u,dv− denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over Dn, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

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Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces

Complex Manifolds ◽

10.1515/coma-2020-0115 ◽

2021 ◽

Vol 8 (1) ◽

pp. 223-229

Author(s):

Callum R. Brodie ◽

Andrei Constantin ◽

Rehan Deen ◽

Andre Lukas

Keyword(s):

Topological Index ◽

Line Bundles ◽

Hirzebruch Surfaces ◽

Del Pezzo

Abstract We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.

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