Cospectral constructions and spectral properties of variations of the distance matrix

Lorenzen, Kate

doi:10.31274/etd-20210609-106

Cited by 1 publication

(1 citation statement)

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“…Properties of other closeness eigenvalues, especially the second largest and the smallest closeness eigenvalues were explored in [17]. Some results on the spectral properties of exponential distance matrix have been obtained in [5,11].…”

Section: Introductionmentioning

confidence: 99%

Spectra of closeness Laplacian and closeness signless Laplacian of graphs

Zheng

Zhou

2022

RAIRO-Oper. Res.

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For a graph $G$ with vertex set $V(G)$ and $u,v\in V(G)$, the distance between vertices $u$ and $v$ in $G$, denoted by $d_G(u,v)$, is the length of a shortest path connecting them and it is $\infty$ if there is no such a path, and the closeness of vertex $u$ in $G$ is $c_G(u)=\sum_{w\in V(G)}2^{-d_G(u,w)}$. Given a graph $G$ that is not necessarily connected, for $u,v\in V(G)$, the closeness matrix of $G$ is the matrix whose $(u,v)$-entry is equal to $2^{-d_G(u,v)}$ if $u\ne v$ and $0$ otherwise, the closeness Laplacian is the matrix whose $(u,v)$-entry is equal to \[ \begin{cases} -2^{-d_G(u,v)} & \mbox{if } u\ne v,\\ c_G(u) & \mbox{otherwise} \end{cases} \] and the closeness signless Laplacian is the matrix whose $(u,v)$-entry is equal to \[ \begin{cases} 2^{-d_G(u,v)} & \mbox{if } u\ne v,\\ c_G(u) & \mbox{otherwise}. \end{cases} \] We establish relations connecting the spectral properties of closeness Laplacian and closeness signless Laplacian and structural properties of graphs. We give tight upper bounds for all nontrivial closeness Laplacian eigenvalues and characterize the extremal graphs, and determine all trees and unicyclic graphs that maximize the second smallest closeness Laplacian eigenvalue. Also, we give tight upper bounds for the closeness signless Laplacian eigenvalues and determine the trees whose largest closeness signless Laplacian eigenvalues achieve the first two largest values.

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Section: Introductionmentioning

confidence: 99%