Energy and inertia of the eccentricity matrix of coalescence of graphs

Patel, Ajay Kumar; Selvaganesh, Lavanya; Pandey, Sanjay Kumar

doi:10.1016/j.disc.2021.112591

Cited by 17 publications

(4 citation statements)

References 15 publications

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“…In [6], Mahato et al computed the inertia of eccentricity matrices of the path and the lollipop graphs. Recently, Patel et al [9] studied the inertia of coalescence of graphs. One of the main objectives of this article is to compute the inertia of eccentricity matrices of trees.…”

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%

“…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. For more details about the E-energy of graphs, we refer to [14,15,18]. Motivated by the above-mentioned works, in this article, we study the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs for the E-spectral radius, second largest E-eigenvalue, least E-eigenvalue and E-energy among the complements of trees.…”

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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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“…In 2020, Wei et al [124] characterized the extremal trees of given diameter having the minimumspectral radius. In 2021, Patel et al [104] studied the irreducibility and the spectrum of the eccentricity matrix for particular classes of graphs, namely windmill graphs, the coalescence of complete graphs, and the coalescence of two cycles. In 2022, Wang et al [121] showed that when the order n tends to infinity, the fractions of non-isomorphic cospectral graphs with respect to the adjacency and the eccentricity matrix behave like those only concerning the self-centered graphs with diameter two.…”

Section: Eccentricity Matrixmentioning

confidence: 99%

Spectral properties of digraphs with a fixed dichromatic number

Yang

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Other papers by the author[1] Ordering of bicyclic signed digraphs by energy, Filomat 34 (2020), 4297-4309 (with L. Wang).[2] Iota energy orderings of bicyclic signed digraphs, Transactions on Combinatorics 10 (2021), 187-200 (with L. Wang).[3] The eccentricity matrix of a digraph, Discrete Applied Mathematics 322 (2022), 61-73 (with L. Wang).[4] On the sum of the k largest absolute values of Laplacian eigenvalues of digraphs,

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Energy and inertia of the eccentricity matrix of coalescence of graphs

Cited by 17 publications

References 15 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

Spectral properties of digraphs with a fixed dichromatic number

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