Article in press
Authors:
Shariefuddin Pirzada
Department of Math., University of Kashmir, Srinagar
Title:
On the distribution of distance signless Laplacian eigenvalues with given independence and chromatic number
PDFSource:
Discussiones Mathematicae Graph Theory
Received: 2022-11-12 , Revised: 2023-09-09 , Accepted: 2023-09-14 , Available online: 2023-10-19 , https://doi.org/10.7151/dmgt.2524
Abstract:
For a connected graph $G$ of order $n$, let $\mathcal {D}(G)$ be the distance
matrix and $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$. The
distance signless Laplacian (dsL, for short) matrix of $G$ is defined as
$\mathcal {D}^Q(G)=Tr(G)+\mathcal{D}(G)$, and the corresponding eigenvalues are
the dsL eigenvalues of $G$. For an interval $I$, let $m_{\mathcal{D}^{Q}(G) } I$
denote the number of dsL eigenvalues of $G$ lying in the interval $I$. In this
paper, for some prescribed interval $I$, we obtain bounds for
$m_{\mathcal{D}^{Q}(G)}I$ in terms of the independence number $\alpha$ and the
chromatic number $\chi$ of $G$. Furthermore, we provide lower bounds of
$\partial_{1}^{Q}(G)$, the dsL spectral radius, for certain families of graphs
in terms of the order $n$ and the independence number $\alpha$, or the chromatic
number $\chi$.
Keywords:
distance matrix, distance signless Laplacian matrix, spectral radius, independence number, chromatic number
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