Spectra of eccentricity matrices of graphs

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

doi:10.1016/j.dam.2020.05.029

Cited by 29 publications

(8 citation statements)

References 4 publications

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“…+ r k , k) as an induced subgraph. Also, from the partition of the matrix, it can contain CS (7,3) or CS (7,4) or CS(n+2, n) as induced subgraph as well as the eccentricity matrices of them is principal submatrix of it. That's, there can be three cases.…”

Section: Theorem 211 Let H Denote the Class Of Connected Graphs Which...mentioning

confidence: 99%

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On the graphs having at most one positive eccentricity eigenvalue

Sorgun¹,

Küçük²

2020

Preprint

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Section: Theorem 211 Let H Denote the Class Of Connected Graphs Which...mentioning

confidence: 99%

“…There are some papers regarding ε-spectra of some standart graphs such as star K 1,k−1 = S k , complete multipartite graph K n 1 ,n 2 ,...n k , complete graph K n etc. (For detail, see [1,2,3]. Also in [10] it is shown that K n and K n 1 ,n 2 ,...,n k are determined by ε-spectrum.…”

Section: Introductionmentioning

confidence: 99%

On the graphs having at most one positive eccentricity eigenvalue

Sorgun¹,

Küçük²

2020

Preprint

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“…They showed that the eccentricity matrix of a tree is irreducible. Mahato et al [16] gave an alternative proof. In [20], this result was generalized by providing a class of graphs whose eccentricity matrices are irreducible.…”

Section: Introductionmentioning

confidence: 99%

“…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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“…Furthermore, Dehmer and Shi [6] studied the uniqueness of D MAX -matrix. Recently, Wang et al [23] studied the graph energy based on the eccentricity matrix; Wang et al [22] studied some spectral properties of the eccentricity matrix of graphs; Mahato et al [15] studied the spectra of graphs based on the eccentricity matrix; Wei et al [20] determined the nvertex trees with minimum ε-spectral radius. Furthermore, in [20], the authors identified all trees with given order and diameter having minimum ε-spectral; Tura et al [19] studied the eccentricity energy of complete multipartite graphs Note that the adjacency matrix A(G) can be regarded as constructed from the distance matrix D(G) by selecting only the smallest distances for each row and each column, which correspond to adjacent vertices.…”

Section: Introductionmentioning

confidence: 99%

On the largest and least eigenvalues of eccentricity matrix of trees

He¹

2020

Preprint

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The eccentricity matrix ε(G) of a graph G is constructed from the distance matrix of G by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The ε-eigenvalues of a graph G are those of its eccentricity matrix ε(G). Wang et al [22] proposed the problem of determining the maximum ε-spectral radius of trees with given order. In this paper, we consider the above problem of n-vertex trees with given diameter. The maximum ε-spectral radius of n-vertex trees with fixed odd diameter is obtained, and the corresponding extremal trees are also determined. The trees with least ε-eigenvalues in [−2 √ 2, 0) have been known. Finally, we determine the trees with least ε-eigenvalues in [−2 − √ 13, −2 √ 2).

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Spectra of eccentricity matrices of graphs

Cited by 29 publications

References 4 publications

On the graphs having at most one positive eccentricity eigenvalue

On the graphs having at most one positive eccentricity eigenvalue

On Eccentricity Matrices of Wheel Graphs

On the largest and least eigenvalues of eccentricity matrix of trees

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