<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μ<sub>n−1</sub> >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2022-0011_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>L</mi><mi>E</mi><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mrow><mo>|</mo> <mrow><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>-</mo><mfrac><mrow><mn>2</mn><mi>m</mi></mrow><mi>n</mi></mfrac></mrow> <mo>|</mo></mrow></mrow></mrow></math>
<tex-math>LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|}</tex-math>
</alternatives>
</inline-formula>, <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2022-0011_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>L</mi><mi>E</mi><mi>L</mi><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msubsup><mrow><msqrt><mrow><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub></mrow></msqrt></mrow></mrow></math>
<tex-math>LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} }</tex-math>
</alternatives>
</inline-formula> and <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2022-0011_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>K</mi><mi>f</mi><mo>=</mo><mi>n</mi><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msubsup><mrow><mfrac><mn>1</mn><mrow><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub></mrow></mfrac></mrow></mrow></math>
<tex-math>Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}}</tex-math>
</alternatives>
</inline-formula> respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.</p>
</abstract>

Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as 


LE=∑i=1n| μi-2mn |
LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|}

, 


LEL=∑i=1n-1μi
LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} }

 and 


Kf=n∑i=1n-11μi
Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}}

 respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.

Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs

Acta Universitatis Sapientiae, Informatica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the...

Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs

Published Online: Feb 04, 2023

Page range: 185 - 198

Received: Oct 01, 2022

Accepted: Nov 06, 2022

DOI: https://doi.org/10.2478/ausi-2022-0011

Keywords
Laplacian matrix, Laplacian energy, Laplacian-energy-like invariant, Kirchhoff index

© 2022 S. Bhatnagar et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs

Published Online: Feb 04, 2023

Page range: 185 - 198

Received: Oct 01, 2022

Accepted: Nov 06, 2022

DOI: https://doi.org/10.2478/ausi-2022-0011

KeywordsLaplacian matrix, Laplacian energy, Laplacian-energy-like invariant, Kirchhoff index

© 2022 S. Bhatnagar et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Laplacian matrix, Laplacian energy, Laplacian-energy-like invariant, Kirchhoff index