Energy of a graph and Randic index

Arizmendi, Gerardo; Arizmendi, Octavio

doi:10.1016/j.laa.2020.09.025

Cited by 20 publications

(12 citation statements)

References 11 publications

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“…The purpose of this work is to define the energy of a vertex associated to the matrices L and L, investigate their properties and relate them to spectral, combinatorial and geometric quantities of the graph. While some of the theorems follow similar developments as the one given for the energy of a vertex [2,3,1], by the geometric properties of L and L, different results and new insights are obtained in this paper for their vertex energies.…”

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confidence: 67%

Laplacian Energies of Vertices

Guerrero¹

2022

Preprint

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In this work, we define the Laplacian and Normalized Laplacian energies of vertices in a graph, we derive some of its properties and relate them to combinatorial, spectral and geometric quantities of the graph. IntroductionNowadays, there are many notions of graph energies defined in the literature (see [12] for a review of them). In the present work we will focus our interest in the Laplacian ([13]) and normalized Laplacian ([6]) energies.We denote a graph by G = (V, E), with n vertices and m edges,is the vertex set, and E is the set of edges. The degree of a vertex v i is the number of vertices to which it is connected, and is denoted by d i . If there is an edge between two vertices v, w ∈ V we write v ∼ w. We will assume that G is simple and connected.The M−energy of a graph G was introduced in [6], where M is a matrix associated to G. The M-energy is defined by

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mentioning

confidence: 67%

Laplacian Energies of Vertices

Guerrero¹

2022

Preprint

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“…In two recently published papers, [57,58] remarkably simple inequalities between the Randić index, Eq. (1), and graph energy, Eq.…”

Section: Randić Index and Graph Energymentioning

confidence: 99%

“…Equality on the left-hand side is attained only if The left inequality was discovered by Yan et al [57] whereas the right inequality by Gerardo and Octavio Arizmendi. [58] In spite of their simplicity, proving the inequalities ( 9) is a very difficult mathematical task. Namely, whereas ( ) R G is a vertex-degree-dependent quantity, ( ) GE G depends on graph eigenvalues.…”

Section: Theorem 12mentioning

confidence: 99%

Some Less Familiar Properties of Randić Index

Gutman

2021

Croat. chem. acta (Online)

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“…Á Á Á ! q n are the signless Laplacian eigenvalues of G and 2m n is the average degree of G. For recent works on the energy, the Laplacian energy and the signless Laplacian energy, we refer to [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

On α-adjacency energy of graphs and Zagreb index

Pirzada

Rather

Ganie³

et al. 2021

AKCE International Journal of Graphs and Combinatorics

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Let A(G) be the adjacency matrix and D(G) be the diagonal matrix of the vertex degrees of a simple connected graph G. Nikiforov defined the matrix A a ðGÞ of the convex combinations of D(G) and A(G) as A a ðGÞ ¼ aDðGÞ þ ð1 À aÞAðGÞ, for 0 a 1: If q 1 ! q 2 ! Á Á Á ! q n are the eigenvalues of A a ðGÞ (which we call a-adjacency eigenvalues of G), the a-adjacency energy of G is defined as E Aa ðGÞ ¼where n is the order and m is the size of G. We obtain upper and lower bounds for E Aa ðGÞ in terms of the order n, the size m and the Zagreb index Zg(G) associated to the structure of G. Further, we characterize the extremal graphs attaining these bounds.

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Energy of a graph and Randic index

Cited by 20 publications

References 11 publications

Laplacian Energies of Vertices

Laplacian Energies of Vertices

Some Less Familiar Properties of Randić Index

On α-adjacency energy of graphs and Zagreb index

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