On the largest and least eigenvalues of eccentricity matrix of trees

He, Xiaocong; Lu, Lu

doi:10.1016/j.disc.2021.112662

Cited by 21 publications

(4 citation statements)

References 18 publications

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“…Wang et al [18] and Mahato et al [11] respectively determined the lower and upper bounds for the ǫ-spectral radius of graphs, and identified the corresponding extremal graphs. He and Lu [4] determined the maximum ǫ-spectral radius of n-vertex trees with fixed odd diameter and characterized the corresponding extremal trees. Yang and Wang [21] characterized the lower bound of the ǫ-spectral radius of a digraph and the corresponding extremal digraphs.…”

Section: Introductionmentioning

confidence: 99%

The ϵ-spectral radii of k-uniform hypertrees *

Liu

2023

Preprint

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The \epsilon-spectral radius of a connected hypergraph is the largest eigenvalue of its eccentricity matrix. Let \mathcal{T}_{m,d} be the set of k-uniform hypertrees with size m and diameter d. In this paper, we show that the eccentricity matrix of a k-uniform hypertree is irreducible, and we characterize the unique hypertrees with the minimum-spectral radius among \bigcup_{d\neq3}\mathcal{T}_{m,d}. AMS Classi cation: 05C50, 05C65

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Section: Introductionmentioning

confidence: 99%

The ϵ-spectral radii of k-uniform hypertrees *

Liu

2023

Preprint

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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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“…Wang et al [19] characterized the graphs with minimum and second minimum E-spectral radius as well as the graphs with maximum least and second least E-eigenvalues. Recently, He and Lu [4] considered the maximal problem for the E-spectral radius of trees with the fixed odd diameter and determined the extremal trees. Wei et al [25] characterized the trees with second minimum E-spectral radius and identified the trees with small matching number having the minimum E-spectral radius.…”

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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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On the largest and least eigenvalues of eccentricity matrix of trees

Cited by 21 publications

References 18 publications

The ϵ-spectral radii of k-uniform hypertrees *

The ϵ-spectral radii of k-uniform hypertrees *

On the eccentricity matrices of trees: Inertia and spectral symmetry

Extremal problems for the eccentricity matrices of complements of trees

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