Positive and negative square energies of graphs

Aida Abiad; Leonardo De Lima; Dheer Noal Desai; Krystal Guo; Leslie Hogben; José Madrid

doi:10.13001/ela.2023.7827

Positive and negative square energies of graphs

Aida Abiad, Leonardo De Lima (Corresponding author), Dheer Noal Desai, Krystal Guo, Leslie Hogben, José Madrid

Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic › peer review

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The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. Let s⁺ (G), s⁻ (G) denote the sum of the squares of the positive and negative eigenvalues of G, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if G is a connected graph of order n, then s⁺ (G) ≥ n − 1 and s⁻ (G) ≥ n − 1. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.

Originele taal-2	Engels
Pagina's (van-tot)	307-326
Aantal pagina's	20
Tijdschrift	Electronic Journal of Linear Algebra
Volume	39
DOI's	https://doi.org/10.13001/ela.2023.7827
Status	Gepubliceerd - mei 2023

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Acknowledgment. This project started and was made possible by Spectral Graph and Hypergraph Theory: Connections & Applications, December 6-10, 2021, a workshop of the American Institute of Math- ematics with support from the US National Science Foundation. The authors thank AIM and also thank Sam Spiro for many fruitful discussions. We thank Clive Elphick for bringing Conjecture 1.1 to Aida Abiad’s attention and also for suggesting Example 4.2. We thank the reviewer for helpful comments and examples that have improved the paper. Aida Abiad is partially supported by the Dutch Research Council through the grant VI.Vidi.213.085 and by the Research Foundation Flanders through the grant 1285921N. Leonardo de Lima is partially supported by CNPq grant 315739/2021-5.

Financiers	Financiernummer
American Institute of Mathematics
National Science Foundation(NSF)
Fonds Wetenschappelijk Onderzoek	1285921N
Nederlandse Organisatie voor Wetenschappelijk Onderzoek	VI.Vidi.213.085
Conselho Nacional de Desenvolvimento Científico e Tecnológico	315739/2021-5

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title = "Positive and negative square energies of graphs",

abstract = "The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. Let s+ (G), s− (G) denote the sum of the squares of the positive and negative eigenvalues of G, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if G is a connected graph of order n, then s+ (G) ≥ n − 1 and s− (G) ≥ n − 1. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.",

keywords = "Adjacency matrix, Energy, Graph eigenvalues, Inertia of a graph",

author = "Aida Abiad and {De Lima}, Leonardo and Desai, {Dheer Noal} and Krystal Guo and Leslie Hogben and Jos{\'e} Madrid",

year = "2023",

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doi = "10.13001/ela.2023.7827",

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AU - Abiad, Aida

AU - De Lima, Leonardo

AU - Desai, Dheer Noal

AU - Guo, Krystal

AU - Hogben, Leslie

AU - Madrid, José

PY - 2023/5

Y1 - 2023/5

N2 - The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. Let s+ (G), s− (G) denote the sum of the squares of the positive and negative eigenvalues of G, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if G is a connected graph of order n, then s+ (G) ≥ n − 1 and s− (G) ≥ n − 1. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.

AB - The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. Let s+ (G), s− (G) denote the sum of the squares of the positive and negative eigenvalues of G, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if G is a connected graph of order n, then s+ (G) ≥ n − 1 and s− (G) ≥ n − 1. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.

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KW - Inertia of a graph

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EP - 326

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JF - Electronic Journal of Linear Algebra

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Positive and negative square energies of graphs

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