Some properties of the distance Laplacian eigenvalues of a graph

Mustapha Aouchiche; Pierre Hansen

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 3, page 751-761
  • ISSN: 0011-4642

Abstract

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The distance Laplacian of a connected graph G is defined by ℒ = Diag ( Tr ) - 𝒟 , where 𝒟 is the distance matrix of G , and Diag ( Tr ) is the diagonal matrix whose main entries are the vertex transmissions in G . The spectrum of ℒ is called the distance Laplacian spectrum of G . In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.

How to cite

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Aouchiche, Mustapha, and Hansen, Pierre. "Some properties of the distance Laplacian eigenvalues of a graph." Czechoslovak Mathematical Journal 64.3 (2014): 751-761. <http://eudml.org/doc/262154>.

@article{Aouchiche2014,
abstract = {The distance Laplacian of a connected graph $G$ is defined by $\mathcal \{L\} = \{\rm Diag(Tr)\}- \mathcal \{D\}$, where $\mathcal \{D\}$ is the distance matrix of $G$, and $\{\rm Diag(Tr)\}$ is the diagonal matrix whose main entries are the vertex transmissions in $G$. The spectrum of $\mathcal \{L\}$ is called the distance Laplacian spectrum of $G$. In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.},
author = {Aouchiche, Mustapha, Hansen, Pierre},
journal = {Czechoslovak Mathematical Journal},
keywords = {distance matrix; Laplacian; characteristic polynomial; eigenvalue; distance matrix; Laplacian; characteristic polynomial; eigenvalue},
language = {eng},
number = {3},
pages = {751-761},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some properties of the distance Laplacian eigenvalues of a graph},
url = {http://eudml.org/doc/262154},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Aouchiche, Mustapha
AU - Hansen, Pierre
TI - Some properties of the distance Laplacian eigenvalues of a graph
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 751
EP - 761
AB - The distance Laplacian of a connected graph $G$ is defined by $\mathcal {L} = {\rm Diag(Tr)}- \mathcal {D}$, where $\mathcal {D}$ is the distance matrix of $G$, and ${\rm Diag(Tr)}$ is the diagonal matrix whose main entries are the vertex transmissions in $G$. The spectrum of $\mathcal {L}$ is called the distance Laplacian spectrum of $G$. In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.
LA - eng
KW - distance matrix; Laplacian; characteristic polynomial; eigenvalue; distance matrix; Laplacian; characteristic polynomial; eigenvalue
UR - http://eudml.org/doc/262154
ER -

References

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