<abstract xmlns="http://www.w3.org/1999/xhtml">Let G be a simple connected graph. The reciprocal transmission Tr′G(ν) of a vertex ν is defined as<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2018-0011_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mrow><mtext>Tr</mtext></mrow></mrow><mtext>G</mtext><mo>′</mo></msubsup><mo>(</mo><mo>ν</mo><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mtext>u</mtext><mo>∈</mo><mtext>V</mtext><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder><mrow><mfrac><mn>1</mn><mrow><msub><mrow><mtext>d</mtext></mrow><mtext>G</mtext></msub><mo>(</mo><mi>u</mi><mo>,</mo><mo>ν</mo><mo>)</mo></mrow></mfrac><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mtext>u</mtext><mo>≠</mo><mo>ν</mo><mo>.</mo></mrow></mrow></math><tex-math>$${\rm{Tr}}_{\rm{G}}^\prime ({\rm{\nu }}) = \sum\limits_{{\rm{u}} \in {\rm{V}}(G)} {{1 \over {{{\rm{d}}_{\rm{G}}}(u,{\rm{\nu }})}}{\rm{u}} \ne {\rm{\nu }}.} $$</tex-math></alternatives></disp-formula>The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph G is defined as RQ(G)= diag(Tr′ (G)) + RD(G), where RD(G) is the Harary matrix (reciprocal distance matrix) of G and diag(Tr′ (G)) is the diagonal matrix of the vertex reciprocal transmissions in G. In this paper, we investigate the RDSL spectrum of some classes of graphs that are arisen from graph operations such as cartesian product, extended double cover product and InduBala product. We introduce minimum covering reciprocal distance signless Laplacian matrix (or briey MCRDSL matrix) of G as the square matrix of order n, RQC(G) := (qi;j),<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2018-0011_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>q</mtext></mrow><mrow><mtext>ij</mtext></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mn>1</mn><mo>+</mo><mtext>Tr</mtext><mo>′</mo><mo>(</mo><msub><mrow><mo>ν</mo></mrow><mtext>i</mtext></msub><mo>)</mo></mrow></mtd><mtd><mrow><mtext>if</mtext></mrow></mtd><mtd><mrow><mtext>i</mtext><mo>=</mo><mtext>j</mtext></mrow></mtd><mtd><mrow><mtext>and</mtext></mrow></mtd><mtd><mrow><msub><mrow><mo>ν</mo></mrow><mtext>i</mtext></msub><mo>∈</mo><mtext>C</mtext></mrow></mtd></mtr><mtr><mtd><mrow><mtext>Tr</mtext><mo>′</mo><mo>(</mo><msub><mrow><mo>ν</mo></mrow><mtext>i</mtext></msub><mo>)</mo></mrow></mtd><mtd><mrow><mtext>if</mtext></mrow></mtd><mtd><mrow><mtext>i</mtext><mo>=</mo><mtext>j</mtext></mrow></mtd><mtd><mrow><mtext>and</mtext></mrow></mtd><mtd><mrow><msub><mrow><mo>ν</mo></mrow><mtext>i</mtext></msub><mo>∉</mo><mtext>C</mtext></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><msub><mrow><mrow><mtext>d</mtext><mo>(</mo><mi>ν</mi></mrow></mrow><mtext>i</mtext></msub><mo>,</mo><msub><mrow><mo>ν</mo></mrow><mtext>j</mtext></msub><mo>)</mo></mrow></mfrac></mrow></mtd><mtd><mrow><mtext>otherwise</mtext></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>$${{\rm{q}}_{{\rm{ij}}}} = \left\{ {\matrix{ {1 + {\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} &amp; {{\rm{if}}} &amp; {{\rm{i = j}}} &amp; {{\rm{and}}} &amp; {{{\rm{\nu }}_{\rm{i}}} \in {\rm{C}}} \cr {{\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} &amp; {{\rm{if}}} &amp; {{\rm{i = j}}} &amp; {{\rm{and}}} &amp; {{{\rm{\nu }}_{\rm{i}}} \notin {\rm{C}}} \cr {{1 \over {{\rm{d(}}{{\rm{\nu }}_{\rm{i}}},{{\rm{\nu }}_{\rm{j}}})}}} &amp; {{\rm{otherwise}}} &amp; {} &amp; {} &amp; {} \cr } } \right.$$</tex-math></alternatives></disp-formula>where C is a minimum vertex cover set of G. MCRDSL energy of a graph G is defined as sum of eigenvalues of RQC. Extremal graphs with respect to MCRDSL energy of graph are characterized. We also obtain some bounds on MCRDSL energy of a graph and MCRDSL spectral radius of 𝒢, which is the largest eigenvalue of the matrix RQC (G) of graphs.</abstract>

Let G be a simple connected graph. The reciprocal transmission Tr′G(ν) of a vertex ν is defined asTrG′(ν)=∑u∈V(G)1dG(u,ν),           u≠ν.$${\rm{Tr}}_{\rm{G}}^\prime ({\rm{\nu }}) = \sum\limits_{{\rm{u}} \in {\rm{V}}(G)} {{1 \over {{{\rm{d}}_{\rm{G}}}(u,{\rm{\nu }})}}{\rm{u}} \ne {\rm{\nu }}.} $$The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph G is defined as RQ(G)= diag(Tr′ (G)) + RD(G), where RD(G) is the Harary matrix (reciprocal distance matrix) of G and diag(Tr′ (G)) is the diagonal matrix of the vertex reciprocal transmissions in G. In this paper, we investigate the RDSL spectrum of some classes of graphs that are arisen from graph operations such as cartesian product, extended double cover product and InduBala product. We introduce minimum covering reciprocal distance signless Laplacian matrix (or briey MCRDSL matrix) of G as the square matrix of order n, RQC(G) := (qi;j),qij={1+Tr′(νi)ifi=jandνi∈CTr′(νi)ifi=jandνi∉C1d(νi,νj)otherwise$${{\rm{q}}_{{\rm{ij}}}} = \left\{ {\matrix{   {1 + {\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \in {\rm{C}}}  \cr    {{\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \notin {\rm{C}}}  \cr    {{1 \over {{\rm{d(}}{{\rm{\nu }}_{\rm{i}}},{{\rm{\nu }}_{\rm{j}}})}}} & {{\rm{otherwise}}} & {} & {} & {}  \cr  } } \right.$$where C is a minimum vertex cover set of G. MCRDSL energy of a graph G is defined as sum of eigenvalues of RQC. Extremal graphs with respect to MCRDSL energy of graph are characterized. We also obtain some bounds on MCRDSL energy of a graph and MCRDSL spectral radius of 𝒢, which is the largest eigenvalue of the matrix RQC (G) of graphs.

Minimum covering reciprocal distance signless Laplacian energy of graphs

Acta Universitatis Sapientiae, Informatica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Minimum covering reciprocal distance signless Laplacian energy of graphs"}

Let G be a simple connected graph. The reciprocal transmission Tr′G(ν) of a vertex ν is defined...

Minimum covering reciprocal distance signless Laplacian energy of graphs

Published Online: Dec 31, 2018

Page range: 218 - 240

Received: Aug 08, 2018

DOI: https://doi.org/10.2478/ausi-2018-0011

Keywords
Reciprocal distance signless Laplacian matrix, reciprocal distance signless Laplacian energy, minimum covering reciprocal distance signless Laplacian energy

© 2018 Abdollah Alhevaz et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Minimum covering reciprocal distance signless Laplacian energy of graphs

Published Online: Dec 31, 2018

Page range: 218 - 240

Received: Aug 08, 2018

DOI: https://doi.org/10.2478/ausi-2018-0011

KeywordsReciprocal distance signless Laplacian matrix, reciprocal distance signless Laplacian energy, minimum covering reciprocal distance signless Laplacian energy

© 2018 Abdollah Alhevaz et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Reciprocal distance signless Laplacian matrix, reciprocal distance signless Laplacian energy, minimum covering reciprocal distance signless Laplacian energy