On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs

Kinkar Ch Das; Kexiang Xu

doi:10.1007/s40840-015-0280-4

On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs

Kinkar Ch Das, Kexiang Xu

Department of Mathematics

Nanjing University of Aeronautics and Astronautics

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

13 Scopus citations

Abstract

Let G= (V, E) be a simple graph of order n with m edges and Laplacian eigenvalues μ₁≥ μ₂≥ ⋯ ≥ μ_n _- ₁≥ μ_n= 0. The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as (Formula presented.) and (Formula presented.) respectively. The Laplacian energy of the graph G is defined as (Formula presented.) In this paper, we present an upper bound on Kf of graphs. Also, we obtain some relations between Kf, LEL and first Zagreb index of G. Finally, we give a relation between LEL and LE of G.

Original language	English
Pages (from-to)	59-75
Number of pages	17
Journal	Bulletin of the Malaysian Mathematical Sciences Society
Volume	39
DOIs	https://doi.org/10.1007/s40840-015-0280-4
State	Published - 1 Jun 2016

Keywords

Graph
Kirchhoff index
Laplacian energy
Laplacian spectrum (of graph)
Laplacian-energy-like invariant
Spanning tree

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10.1007/s40840-015-0280-4

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title = "On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs",

abstract = "Let G= (V, E) be a simple graph of order n with m edges and Laplacian eigenvalues μ1≥ μ2≥ ⋯ ≥ μn - 1≥ μn= 0. The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as (Formula presented.) and (Formula presented.) respectively. The Laplacian energy of the graph G is defined as (Formula presented.) In this paper, we present an upper bound on Kf of graphs. Also, we obtain some relations between Kf, LEL and first Zagreb index of G. Finally, we give a relation between LEL and LE of G.",

keywords = "Graph, Kirchhoff index, Laplacian energy, Laplacian spectrum (of graph), Laplacian-energy-like invariant, Spanning tree",

author = "Das, \{Kinkar Ch\} and Kexiang Xu",

note = "Publisher Copyright: {\textcopyright} 2015, Malaysian Mathematical Sciences Society and Universiti Sains Malaysia.",

year = "2016",

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doi = "10.1007/s40840-015-0280-4",

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T1 - On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs

AU - Das, Kinkar Ch

AU - Xu, Kexiang

PY - 2016/6/1

Y1 - 2016/6/1

N2 - Let G= (V, E) be a simple graph of order n with m edges and Laplacian eigenvalues μ1≥ μ2≥ ⋯ ≥ μn - 1≥ μn= 0. The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as (Formula presented.) and (Formula presented.) respectively. The Laplacian energy of the graph G is defined as (Formula presented.) In this paper, we present an upper bound on Kf of graphs. Also, we obtain some relations between Kf, LEL and first Zagreb index of G. Finally, we give a relation between LEL and LE of G.

AB - Let G= (V, E) be a simple graph of order n with m edges and Laplacian eigenvalues μ1≥ μ2≥ ⋯ ≥ μn - 1≥ μn= 0. The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as (Formula presented.) and (Formula presented.) respectively. The Laplacian energy of the graph G is defined as (Formula presented.) In this paper, we present an upper bound on Kf of graphs. Also, we obtain some relations between Kf, LEL and first Zagreb index of G. Finally, we give a relation between LEL and LE of G.

KW - Graph

KW - Kirchhoff index

KW - Laplacian energy

KW - Laplacian spectrum (of graph)

KW - Laplacian-energy-like invariant

KW - Spanning tree

UR - https://www.scopus.com/pages/publications/84976322335

U2 - 10.1007/s40840-015-0280-4

DO - 10.1007/s40840-015-0280-4

M3 - Article

AN - SCOPUS:84976322335

SN - 0126-6705

VL - 39

SP - 59

EP - 75

JO - Bulletin of the Malaysian Mathematical Sciences Society

JF - Bulletin of the Malaysian Mathematical Sciences Society

ER -

On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs

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