Degree-based energies of graphs

Let G = (V, E), V = {1, 2,. .. , n}, be a simple graph of order n and size m, without isolated vertices. Denote by ∆ = d 1 ≥ d 2 ≥ • • • ≥ d n = δ > 0, d i = d(i), a sequence of its vertex degrees. If vertices i and j are adjacent, we write i ∼ j. With T I we denote a topological index that can be represented as T I = T I(G) = ∑ i∼ j F(d i , d j), where F is an appropriately chosen function with the property F(x, y) = F(y, x). Randić degree-based adjacency matrix RA = (r i j) is defined as r i j = F(d i ,d j) √ d i d j if i ∼ j, and 0 otherwise. Denote by f i , i = 1, 2,. .. , n, the eigenvalues of RA. The Randić degree-based energy of graph could be defined as RE T I = RE T I (G) = ∑ n i=1 | f i |. Upper and lower bounds for RE T I are obtained.

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“…The inequality (18) was proven in [4]. Since, in this case, f 1 − f n ≤ 2, this inequality is stronger then RE ≥ 2R −1 , which was proved in [3].…”

Section: Preliminariesmentioning

confidence: 90%

Randić degree-based energy of graphs

Zogić¹,

Borovićanin²,

Milovanovic³

et al. 2018

Sci Pub Univ Novi Pazar Ser A

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“…Das et al [8] prove the following theorem for eigenvalues of degree based energies of graphs. [8]). For the eigenvalues f 1 ≥ f 2 ≥ · · · ≥ f n of a matrix M, the followimg inequalities hold.…”

Section: Bounds and Integral Representation For Isi Energymentioning

confidence: 99%

Inverse Sum Indeg Energy of Graphs

Hafeez

Farooq

2019

IEEE Access

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Suppose G is an n-vertex simple graph with vertex set {v 1 , . . . , v n } andif the vertices v i and v j are adjacent and s ij = 0 otherwise. The S-eigenvalues of G are the eigenvalues of its ISI matrix S(G). Recently the notion of inverse sum indeg (henceforth, ISI) energy of graphs is introduced and is defined by n i=1 |τ i |, where τ i are the S-eigenvalues. We give ISI energy formula of some graph classes. We also obtain some bounds for ISI energy of graphs.

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“…The paper [21] studied the spectrum and energy of arithmetic-geometric matrix, in which the sum of all elements is equal to 2AG. Other bounds of the arithmetic-geometric energy appeared in [22,23]. The paper [24] studies extremal AG-graphs for various classes of simple graphs, and it includes inequalities that involve AG + GA, AG − GA, AG • GA, and AG/GA.…”

Section: Introductionmentioning

confidence: 99%

Bounds on the Arithmetic-Geometric Index

et al. 2021

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The concept of arithmetic-geometric index was recently introduced in chemical graph theory, but it has proven to be useful from both a theoretical and practical point of view. The aim of this paper is to obtain new bounds of the arithmetic-geometric index and characterize the extremal graphs with respect to them. Several bounds are based on other indices, such as the second variable Zagreb index or the general atom-bond connectivity index), and some of them involve some parameters, such as the number of edges, the maximum degree, or the minimum degree of the graph. In most bounds, the graphs for which equality is attained are regular or biregular, or star graphs.

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Degree-based energies of graphs

Cited by 57 publications

References 23 publications

Randić degree-based energy of graphs

Randić degree-based energy of graphs

Inverse Sum Indeg Energy of Graphs

Bounds on the Arithmetic-Geometric Index

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