On the eigenvalues of eccentricity matrix of graphs

Lei, Xingyu; Wang, Jianfeng; Li, Guozheng

doi:10.1016/j.dam.2021.02.029

Cited by 18 publications

(3 citation statements)

References 11 publications

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“…Wei and Li [17] established a relationship between the majorization and E-spectral radii of complete multipartite graphs. For more details about the eccentricity matrices of graphs,we refer to [2,4,5,10,11,12,13,14,18].…”

Section: Introductionmentioning

confidence: 99%

On the eccentricity matrices of trees: Inertia and spectral symmetry

Mahato¹,

Kannan²

2022

Preprint

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The eccentricity matrix E(G) of a connected graph G is obtained from the distance matrix of G by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of E(G) are the E-eigenvalues of G. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than 4 vertices with odd diameter has two positive and two negative E-eigenvalues (irrespective of the structure of the tree). A tree with even diameter has the same number of positive and negative E-eigenvalues, which is equal to the number of 'diametrically distinguished' vertices (see Definition 3.1). Besides we prove that the spectrum of the eccentricity matrix of a tree is symmetric with respect to the origin if and only if the tree has odd diameter. As an application, we characterize the trees with three distinct E-eigenvalues.

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Section: Introductionmentioning

confidence: 99%

On the eccentricity matrices of trees: Inertia and spectral symmetry

Mahato¹,

Kannan²

2022

Preprint

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“…Mahato and Kannan [16] considered the extremal problem for the second largest E-eigenvalue of trees and determined the unique tree with minimum second largest E-eigenvalue among all trees on n vertices other than the star. For more advances on the eccentricity matrices of graphs, we refer to [7,8,13,15,17,18,21,22].…”

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confidence: 99%

“…In [20], Wang et al studied the E-energy of graphs and obtained some bounds for the E-energy of graphs and determined the corresponding extremal graphs. Lei et al [8] obtained an upper bound for the E-energy of graphs and characterized the extremal graphs. Very recently, Mahato and Kannan [16] studied the minimization problem for the E-energy of trees and characterized the trees with minimum E-energy among all trees on n vertices.…”

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confidence: 99%

Extremal problems for the eccentricity matrices of complements of trees

Mahato

Kannan

2023

ELA

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The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$. The largest modulus of an eigenvalue is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) $\mathcal{E}$-spectral radius among the complements of trees. Then, we prove that the $\mathcal{E}$-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least $\mathcal{E}$-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest $\mathcal{E}$-eigenvalue and the $\mathcal{E}$-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum-type lower bounds for the second largest $\mathcal{E}$-eigenvalue and $\mathcal{E}$-energy of a tree and its complement.

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The Complete Classification of Graphs whose Second Largest Eigenvalue of the Eccentricity Matrix is Less Than 1

Wang,

Lei,

et al. 2024

Acta. Math. Sin.-English Ser.

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On the eigenvalues of eccentricity matrix of graphs

Cited by 18 publications

References 11 publications

On the eccentricity matrices of trees: Inertia and spectral symmetry

On the eccentricity matrices of trees: Inertia and spectral symmetry

Extremal problems for the eccentricity matrices of complements of trees

The Complete Classification of Graphs whose Second Largest Eigenvalue of the Eccentricity Matrix is Less Than 1

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