On Laplacian eigenvalues of connected graphs

Igor Ž. Milovanović; Emina I. Milovanović; Edin Glogić

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 529-535
  • ISSN: 0011-4642

Abstract

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Let G be an undirected connected graph with n , n ≥ 3 , vertices and m edges with Laplacian eigenvalues μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n - 1 > μ n = 0 . Denote by μ I = μ r 1 + μ r 2 + ⋯ + μ r k , 1 ≤ k ≤ n - 2 , 1 ≤ r 1 < r 2 < ⋯ < r k ≤ n - 1 , the sum of k arbitrary Laplacian eigenvalues, with μ I 1 = μ 1 + μ 2 + ⋯ + μ k and μ I n = μ n - k + ⋯ + μ n - 1 . Lower bounds of graph invariants μ I 1 - μ I n and μ I 1 / μ I n are obtained. Some known inequalities follow as a special case.

How to cite

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Milovanović, Igor Ž., Milovanović, Emina I., and Glogić, Edin. "On Laplacian eigenvalues of connected graphs." Czechoslovak Mathematical Journal 65.2 (2015): 529-535. <http://eudml.org/doc/270127>.

@article{Milovanović2015,
abstract = {Let $G$ be an undirected connected graph with $n$, $n\ge 3$, vertices and $m$ edges with Laplacian eigenvalues $\mu _1\ge \mu _2 \ge \cdots \ge \mu _\{n-1\}>\mu _n =0$. Denote by $\mu _I =\mu _\{r_1\}+\mu _\{r_2\} +\cdots +\mu _\{r_k\}$, $1\le k\le n-2$, $1\le r_1<r_2<\cdots <r_k\le n-1$, the sum of $k$ arbitrary Laplacian eigenvalues, with $\mu _\{I_1\}=\mu _1+\mu _2+\cdots +\mu _k$ and $\mu _\{I_n\}=\mu _\{n-k\}+\cdots +\mu _\{n-1\}$. Lower bounds of graph invariants $\mu _\{I_1\}-\mu _\{I_n\}$ and $\{\mu _\{I_1\}\}/\{\mu _\{I_n\}\}$ are obtained. Some known inequalities follow as a special case.},
author = {Milovanović, Igor Ž., Milovanović, Emina I., Glogić, Edin},
journal = {Czechoslovak Mathematical Journal},
keywords = {Laplacian eigenvalues; linear spread; ratio spread},
language = {eng},
number = {2},
pages = {529-535},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Laplacian eigenvalues of connected graphs},
url = {http://eudml.org/doc/270127},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Milovanović, Igor Ž.
AU - Milovanović, Emina I.
AU - Glogić, Edin
TI - On Laplacian eigenvalues of connected graphs
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 529
EP - 535
AB - Let $G$ be an undirected connected graph with $n$, $n\ge 3$, vertices and $m$ edges with Laplacian eigenvalues $\mu _1\ge \mu _2 \ge \cdots \ge \mu _{n-1}>\mu _n =0$. Denote by $\mu _I =\mu _{r_1}+\mu _{r_2} +\cdots +\mu _{r_k}$, $1\le k\le n-2$, $1\le r_1<r_2<\cdots <r_k\le n-1$, the sum of $k$ arbitrary Laplacian eigenvalues, with $\mu _{I_1}=\mu _1+\mu _2+\cdots +\mu _k$ and $\mu _{I_n}=\mu _{n-k}+\cdots +\mu _{n-1}$. Lower bounds of graph invariants $\mu _{I_1}-\mu _{I_n}$ and ${\mu _{I_1}}/{\mu _{I_n}}$ are obtained. Some known inequalities follow as a special case.
LA - eng
KW - Laplacian eigenvalues; linear spread; ratio spread
UR - http://eudml.org/doc/270127
ER -

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