Relations between Kirchhoff index and Laplacian-energy-like invariant

B. Arsić; I. Gutman; K. Ch Das; K. Xu

Relations between Kirchhoff index and Laplacian-energy-like invariant

B. Arsić, I. Gutman, K. Ch Das, K. Xu

Department of Mathematics

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

9 Scopus citations

Abstract

The Kirchhoff index Kf and the Laplacian-energy- like invariant LEL are two graph invariants defined in terms of the Laplacian eigenvalues. If μ₁ > μ₂ > . . . > μ_n-1 > μ_n = 0 are the Laplacian eigenvalues of a connected n-vertex graph, then Kf = n σ^n-1 _i=1 1/μ_i and LEL = σ^n-1 _i=1 √ μ_i. We examine the conditions under which Kf > LEL. Among other results we show that Kf > LEL holds for all trees, unicyclic, bicyclic, tricyclic, and tetracyclic connected graphs, except for a finite number of graphs. These exceptional graphs are determined.

Original language	English
Pages (from-to)	59-70
Number of pages	12
Journal	Bulletin, Classe des Sciences Mathematiques et Naturelles, Sciences Mathematiques
Volume	144
Issue number	37
State	Published - 2012

Keywords

Kirchhoff index
Laplacian eigenvalue
Laplacian spectrum (of graph)
Laplacian-energy-like invariant
LEL

Cite this

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title = "Relations between Kirchhoff index and Laplacian-energy-like invariant",

abstract = "The Kirchhoff index Kf and the Laplacian-energy- like invariant LEL are two graph invariants defined in terms of the Laplacian eigenvalues. If μ1 > μ2 > . . . > μn-1 > μn = 0 are the Laplacian eigenvalues of a connected n-vertex graph, then Kf = n σn-1 i=1 1/μi and LEL = σn-1 i=1 √ μi. We examine the conditions under which Kf > LEL. Among other results we show that Kf > LEL holds for all trees, unicyclic, bicyclic, tricyclic, and tetracyclic connected graphs, except for a finite number of graphs. These exceptional graphs are determined.",

keywords = "Kirchhoff index, Laplacian eigenvalue, Laplacian spectrum (of graph), Laplacian-energy-like invariant, LEL",

author = "B. Arsi{\'c} and I. Gutman and Das, \{K. Ch\} and K. Xu",

year = "2012",

language = "English",

volume = "144",

pages = "59--70",

journal = "Bulletin, Classe des Sciences Mathematiques et Naturelles, Sciences Mathematiques",

issn = "0561-7332",

publisher = "Serbian Academy of Sciences and Arts",

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TY - JOUR

T1 - Relations between Kirchhoff index and Laplacian-energy-like invariant

AU - Arsić, B.

AU - Gutman, I.

AU - Das, K. Ch

AU - Xu, K.

PY - 2012

Y1 - 2012

N2 - The Kirchhoff index Kf and the Laplacian-energy- like invariant LEL are two graph invariants defined in terms of the Laplacian eigenvalues. If μ1 > μ2 > . . . > μn-1 > μn = 0 are the Laplacian eigenvalues of a connected n-vertex graph, then Kf = n σn-1 i=1 1/μi and LEL = σn-1 i=1 √ μi. We examine the conditions under which Kf > LEL. Among other results we show that Kf > LEL holds for all trees, unicyclic, bicyclic, tricyclic, and tetracyclic connected graphs, except for a finite number of graphs. These exceptional graphs are determined.

AB - The Kirchhoff index Kf and the Laplacian-energy- like invariant LEL are two graph invariants defined in terms of the Laplacian eigenvalues. If μ1 > μ2 > . . . > μn-1 > μn = 0 are the Laplacian eigenvalues of a connected n-vertex graph, then Kf = n σn-1 i=1 1/μi and LEL = σn-1 i=1 √ μi. We examine the conditions under which Kf > LEL. Among other results we show that Kf > LEL holds for all trees, unicyclic, bicyclic, tricyclic, and tetracyclic connected graphs, except for a finite number of graphs. These exceptional graphs are determined.

KW - Kirchhoff index

KW - Laplacian eigenvalue

KW - Laplacian spectrum (of graph)

KW - Laplacian-energy-like invariant

KW - LEL

UR - https://www.scopus.com/pages/publications/84881627853

M3 - Article

AN - SCOPUS:84881627853

SN - 0561-7332

VL - 144

SP - 59

EP - 70

JO - Bulletin, Classe des Sciences Mathematiques et Naturelles, Sciences Mathematiques

JF - Bulletin, Classe des Sciences Mathematiques et Naturelles, Sciences Mathematiques

IS - 37

ER -

Relations between Kirchhoff index and Laplacian-energy-like invariant

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Keywords

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