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

For a simple connected graph G of order n having distance signless Laplacian eigenvalues <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2021-0028_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>ρ</mi></mrow><mn>1</mn><mi>Q</mi></msubsup><mo>≥</mo><msubsup><mrow><mi>ρ</mi></mrow><mn>2</mn><mi>Q</mi></msubsup><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msubsup><mrow><mi>ρ</mi></mrow><mi>n</mi><mi>Q</mi></msubsup></mrow></math>
<tex-math>\rho _1^Q \ge \rho _2^Q \ge \cdots \ge \rho _n^Q</tex-math>
</alternatives>
</inline-formula>, the distance signless Laplacian energy DSLE(G) is defined as <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2021-0028_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>D</mi><mi>S</mi><mi>L</mi><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mrow><mo>|</mo> <mrow><msubsup><mrow><mi>ρ</mi></mrow><mi>i</mi><mi>Q</mi></msubsup><mo>-</mo><mfrac><mrow><mn>2</mn><mi>W</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mi>n</mi></mfrac></mrow> <mo>|</mo></mrow></mrow></mrow></math>
<tex-math>DSLE\left( G \right) = \sum\nolimits_{i = 1}^n {\left| {\rho _i^Q - {{2W\left( G \right)} \over n}} \right|}</tex-math>
</alternatives>
</inline-formula> where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph Kk ∨ ( Kt∪ Kn−k−t),<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2021-0028_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mrow><mo>⌊</mo> <mrow><mfrac><mrow><mi>n</mi><mo>-</mo><mi>k</mi></mrow><mn>2</mn></mfrac></mrow> <mo>⌋</mo></mrow></mrow></math>
<tex-math>1 \le t \le \left\lfloor {{{n - k} \over 2}} \right\rfloor</tex-math>
</alternatives>
</inline-formula> has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.
</abstract>

For a simple connected graph G of order n having distance signless Laplacian eigenvalues 


ρ1Q≥ρ2Q≥⋯≥ρnQ
\rho _1^Q \ge \rho _2^Q \ge  \cdots  \ge \rho _n^Q

, the distance signless Laplacian energy DSLE(G) is defined as 


DSLE(G)=∑i=1n| ρiQ-2W(G)n |
DSLE\left( G \right) = \sum\nolimits_{i = 1}^n {\left| {\rho _i^Q - {{2W\left( G \right)} \over n}} \right|}

 where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph Kk ∨ ( Kt∪ Kn−k−t),


1≤t≤⌊ n-k2 ⌋
1 \le t \le \left\lfloor {{{n - k} \over 2}} \right\rfloor

 has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.

On graphs with minimal distance signless Laplacian energy

Acta Universitatis Sapientiae, Mathematica

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

{"article-title":"On graphs with minimal distance signless Laplacian energy"}

For a simple connected graph G of order n having distance signless Laplacian eigenvalues 


ρ1Q≥ρ2Q≥⋯≥ρnQ
\rho _1^Q \ge \rho _2^Q \ge  \cdots  \ge \rho _n^Q

,...

On graphs with minimal distance signless Laplacian energy

Published Online: Dec 30, 2021

Page range: 450 - 467

DOI: https://doi.org/10.2478/ausm-2021-0028

Keywords
distance signless Laplacian matrix, independence number, vertex connectivity, extremal graphs

© 2021 S. Pirzada et al., published by Sciendo

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

On graphs with minimal distance signless Laplacian energy

Published Online: Dec 30, 2021

Page range: 450 - 467

DOI: https://doi.org/10.2478/ausm-2021-0028

Keywordsdistance signless Laplacian matrix, independence number, vertex connectivity, extremal graphs

© 2021 S. Pirzada et al., published by Sciendo

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

Keywords
distance signless Laplacian matrix, independence number, vertex connectivity, extremal graphs