Open problem on σ-invariant

Kinkar Ch Das; Seyed Ahmad Mojallal

doi:10.11650/tjm/181104

Open problem on σ-invariant

Kinkar Ch Das
, Seyed Ahmad Mojallal

Department of Mathematics

Sungkyunkwan University

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Let G be a graph of order n with m edges. Also let µ₁ ≥ µ₂ ≥ . . . ≥ µ_n-1 ≥ µ_n = 0 be the Laplacian eigenvalues of graph G and let σ = σ(G) (1 ≤ σ ≤ n) be the largest positive integer such that µ_σ ≥ 2m=n. In this paper, we prove that µ₂(G) ≥ 2m=n for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with σ = 1.

Original language	English
Pages (from-to)	1041-1059
Number of pages	19
Journal	Taiwanese Journal of Mathematics
Volume	23
Issue number	5
DOIs	https://doi.org/10.11650/tjm/181104
State	Published - Oct 2019

Keywords

Average degree
Graph
Laplacian energy
Laplacian matrix
Second largest laplacian eigenvalue
σ-invariant

Access to Document

10.11650/tjm/181104

Cite this

@article{bedce2027f754bd7bfa3eee39ba52020,

title = "Open problem on σ-invariant",

abstract = "Let G be a graph of order n with m edges. Also let µ1 ≥ µ2 ≥ . . . ≥ µn-1 ≥ µn = 0 be the Laplacian eigenvalues of graph G and let σ = σ(G) (1 ≤ σ ≤ n) be the largest positive integer such that µσ ≥ 2m=n. In this paper, we prove that µ2(G) ≥ 2m=n for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with σ = 1.",

keywords = "Average degree, Graph, Laplacian energy, Laplacian matrix, Second largest laplacian eigenvalue, σ-invariant",

author = "Das, \{Kinkar Ch\} and Mojallal, \{Seyed Ahmad\}",

year = "2019",

month = oct,

doi = "10.11650/tjm/181104",

language = "English",

volume = "23",

pages = "1041--1059",

journal = "Taiwanese Journal of Mathematics",

issn = "1027-5487",

publisher = "Mathematical Society of the Rep. of China",

number = "5",

}

TY - JOUR

T1 - Open problem on σ-invariant

AU - Das, Kinkar Ch

AU - Mojallal, Seyed Ahmad

PY - 2019/10

Y1 - 2019/10

N2 - Let G be a graph of order n with m edges. Also let µ1 ≥ µ2 ≥ . . . ≥ µn-1 ≥ µn = 0 be the Laplacian eigenvalues of graph G and let σ = σ(G) (1 ≤ σ ≤ n) be the largest positive integer such that µσ ≥ 2m=n. In this paper, we prove that µ2(G) ≥ 2m=n for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with σ = 1.

AB - Let G be a graph of order n with m edges. Also let µ1 ≥ µ2 ≥ . . . ≥ µn-1 ≥ µn = 0 be the Laplacian eigenvalues of graph G and let σ = σ(G) (1 ≤ σ ≤ n) be the largest positive integer such that µσ ≥ 2m=n. In this paper, we prove that µ2(G) ≥ 2m=n for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with σ = 1.

KW - Average degree

KW - Graph

KW - Laplacian energy

KW - Laplacian matrix

KW - Second largest laplacian eigenvalue

KW - σ-invariant

UR - https://www.scopus.com/pages/publications/85073465568

U2 - 10.11650/tjm/181104

DO - 10.11650/tjm/181104

M3 - Article

AN - SCOPUS:85073465568

SN - 1027-5487

VL - 23

SP - 1041

EP - 1059

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

IS - 5

ER -

Open problem on σ-invariant

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this