On the signless Laplacian and normalized signless Laplacian spreads of graphs

Emina Milovanović; Serife B. Bozkurt Altindağ; Marjan Matejić; Igor Milovanović

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 2, page 499-511
  • ISSN: 0011-4642

Abstract

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Let G = ( V , E ) , V = { v 1 , v 2 , ... , v n } , be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d 1 ≥ d 2 ≥ ⋯ ≥ d n . Denote by A and D the adjacency matrix and diagonal vertex degree matrix of G , respectively. The signless Laplacian of G is defined as L + = D + A and the normalized signless Laplacian matrix as ℒ + = D - 1 / 2 L + D - 1 / 2 . The normalized signless Laplacian spreads of a connected nonbipartite graph G are defined as r ( G ) = γ 2 + / γ n + and l ( G ) = γ 2 + - γ n + , where γ 1 + ≥ γ 2 + ≥ ⋯ ≥ γ n + ≥ 0 are eigenvalues of ℒ + . We establish sharp lower and upper bounds for the normalized signless Laplacian spreads of connected graphs. In addition, we present a better lower bound on the signless Laplacian spread.

How to cite

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Milovanović, Emina, et al. "On the signless Laplacian and normalized signless Laplacian spreads of graphs." Czechoslovak Mathematical Journal 73.2 (2023): 499-511. <http://eudml.org/doc/299397>.

@article{Milovanović2023,
abstract = {Let $G=(V,E)$, $V=\lbrace v_1,v_2,\ldots ,v_n\rbrace $, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^+=D+A$ and the normalized signless Laplacian matrix as $\mathcal \{L\}^+=D^\{-1/2\}L^+ D^\{-1/2\}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r(G)= \gamma _\{2\}^\{+\}/ \gamma _\{n\}^\{+\}$ and $l(G)=\gamma _\{2\}^\{+\}-\gamma _\{n\}^\{+\}$, where $\gamma _1^+ \ge \gamma _2^+\ge \cdots \ge \gamma _n^+ \ge 0$ are eigenvalues of $\mathcal \{L\}^+$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads of connected graphs. In addition, we present a better lower bound on the signless Laplacian spread.},
author = {Milovanović, Emina, Bozkurt Altindağ, Serife B., Matejić, Marjan, Milovanović, Igor},
journal = {Czechoslovak Mathematical Journal},
keywords = {Laplacian graph spectra; bipartite graph; spread of graph},
language = {eng},
number = {2},
pages = {499-511},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the signless Laplacian and normalized signless Laplacian spreads of graphs},
url = {http://eudml.org/doc/299397},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Milovanović, Emina
AU - Bozkurt Altindağ, Serife B.
AU - Matejić, Marjan
AU - Milovanović, Igor
TI - On the signless Laplacian and normalized signless Laplacian spreads of graphs
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 499
EP - 511
AB - Let $G=(V,E)$, $V=\lbrace v_1,v_2,\ldots ,v_n\rbrace $, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^+=D+A$ and the normalized signless Laplacian matrix as $\mathcal {L}^+=D^{-1/2}L^+ D^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r(G)= \gamma _{2}^{+}/ \gamma _{n}^{+}$ and $l(G)=\gamma _{2}^{+}-\gamma _{n}^{+}$, where $\gamma _1^+ \ge \gamma _2^+\ge \cdots \ge \gamma _n^+ \ge 0$ are eigenvalues of $\mathcal {L}^+$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads of connected graphs. In addition, we present a better lower bound on the signless Laplacian spread.
LA - eng
KW - Laplacian graph spectra; bipartite graph; spread of graph
UR - http://eudml.org/doc/299397
ER -

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