On the spectral radius and the energy of eccentricity matrices of graphs

Mahato, Iswar; Gurusamy, R.; Kannan, M. Rajesh; Arockiaraj, S.

doi:10.1080/03081087.2021.2015274

Cited by 21 publications

(6 citation statements)

References 13 publications

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“…Therefore, rank(D(T )) = n for any tree T on n vertices. In [7], the authors proved that the star K 1,n−1 is the only tree of order n for which the eccentricity matrix is invertible, that is, rank…”

Section: Inertia Of Eccentricity Matrices Of Treesmentioning

confidence: 99%

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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: Inertia Of Eccentricity Matrices Of Treesmentioning

confidence: 99%

“…The distance matrix of a tree is always invertible. However, the eccentricity matrix of a tree need not be invertible [7].…”

Section: Introductionmentioning

confidence: 99%

On the eccentricity matrices of trees: Inertia and spectral symmetry

Mahato¹,

Kannan²

2022

Preprint

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“…Wei, He and Li [20] determined the trees with given diameter or fixed order minimizing the ǫspectral radius. 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.…”

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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“…For more details on the applications of E(G) in terms of molecular descriptor, we refer to [19,21,18]. Motivated by the concepts and results of other graph matrices, several spectral properties have been studied for the eccentricity matrix in [19,20,21,16,17,18]. The relation between the eigenvalues of A(G) and E(G) has been investigated for certain graphs in [19].…”

Section: Introductionmentioning

confidence: 99%

On Eccentricity Matrices of Wheel Graphs

Jeyaraman¹,

Divyadevi²

2022

Preprint

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The eccentricity matrix E(G) of a simple connected graph G is obtained from the distance matrix D(G) of G by retaining the largest distance in each row and column, and by defining the remaining entries to be zero. This paper focuses on the eccentricity matrix E(W n ) of the wheel graph W n with n vertices. By establishing a formula for the determinant of E(W n ), we show that E(W n ) is invertible if and only if n ≡ 1 (mod 3). We derive a formula for the inverse of E(W n ) by finding a vector w ∈ R n and an n × n symmetric Laplacian-like matrix L of rank n − 1 such thatFurther, we prove an analogous result for the Moore-Penrose inverse of E(W n ) for the singular case. We also determine the inertia of E(W n ).

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On the spectral radius and the energy of eccentricity matrices of graphs

Cited by 21 publications

References 13 publications

On the eccentricity matrices of trees: Inertia and spectral symmetry

On the eccentricity matrices of trees: Inertia and spectral symmetry

The ϵ-spectral radii of k-uniform hypertrees *

On Eccentricity Matrices of Wheel Graphs

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