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

<p>The transmission of a vertex u in a connected graph G is defined as σ(u) = Σ<sub>v∈V(G)</sub> d(u, v) and reciprocal transmission of a vertex u is defined as <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2022-0006_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>r</mi><mi>s</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><mrow><mfrac><mn>1</mn><mrow><mi>d</mi><mrow><mo>(</mo><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mrow></math>
<tex-math>rs(u) = \sum\nolimits_{v \in V\left( G \right)} {{1 \over {d\left( {u,v} \right)}}}</tex-math>
</alternatives>
</inline-formula>, where d(u, v) is the distance between vertex u and v in G. In this paper we define new distance based topological index of a connected graph G called transmission-reciprocal transmission index <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2022-0006_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>T</mi><mi>R</mi><mi>T</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>r</mi><mi>s</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><mrow><mi>r</mi><mi>s</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></math>
<tex-math>TRT\left( G \right) = \sum\nolimits_{uv \in E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)}</tex-math>
</alternatives>
</inline-formula> and its coindex <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2022-0006_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>T</mi><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∉</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>r</mi><mi>s</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><mrow><mi>r</mi><mi>s</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></math>
<tex-math>\overline {TRT} \left( G \right) = \sum\nolimits_{uv \notin E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)}</tex-math>
</alternatives>
</inline-formula>, where E(G) is the edge set of a graph G and establish the relation between TRT(G) and <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2022-0006_eq_004.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>T</mi><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>
<tex-math>\overline {TRT} \left( G \right)</tex-math>
</alternatives>
</inline-formula>(G). Further compute this index for some standard class of graphs and obtain bounds for it.</p>
</abstract>

The transmission of a vertex u in a connected graph G is defined as σ(u) = Σv∈V(G) d(u, v) and reciprocal transmission of a vertex u is defined as 


rs(u)=∑v∈V(G)1d(u,v)
rs(u) = \sum\nolimits_{v \in V\left( G \right)} {{1 \over {d\left( {u,v} \right)}}}

, where d(u, v) is the distance between vertex u and v in G. In this paper we define new distance based topological index of a connected graph G called transmission-reciprocal transmission index 


TRT(G)=∑uv∈E(G)(σ(u)rs(u)+σ(v)rs(v))
TRT\left( G \right) = \sum\nolimits_{uv \in E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)}

 and its coindex 


TRT¯(G)=∑uv∉E(G)(σ(u)rs(u)+σ(v)rs(v))
\overline {TRT} \left( G \right) = \sum\nolimits_{uv \notin E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)}

, where E(G) is the edge set of a graph G and establish the relation between TRT(G) and 


TRT¯(G)
\overline {TRT} \left( G \right)

(G). Further compute this index for some standard class of graphs and obtain bounds for it.

Transmission-reciprocal transmission index and coindex of graphs

Acta Universitatis Sapientiae, Informatica

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

{"article-title":"Transmission-reciprocal transmission index and coindex of graphs"}

The transmission of a vertex u in a connected graph G is defined as σ(u) = Σv∈V(G) d(u, v) and reciprocal transmission of a vertex u is defined as...

Transmission-reciprocal transmission index and coindex of graphs

Published Online: Aug 12, 2022

Page range: 84 - 103

Received: Apr 27, 2022

Accepted: May 18, 2022

DOI: https://doi.org/10.2478/ausi-2022-0006

Keywords
Distance, transmission of a vertex, reciprocal transmission of a vertex, transmission-reciprocal transmission index

© 2022 Harishchandra S. Ramane et al., published by Sciendo

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

Transmission-reciprocal transmission index and coindex of graphs

Published Online: Aug 12, 2022

Page range: 84 - 103

Received: Apr 27, 2022

Accepted: May 18, 2022

DOI: https://doi.org/10.2478/ausi-2022-0006

KeywordsDistance, transmission of a vertex, reciprocal transmission of a vertex, transmission-reciprocal transmission index

© 2022 Harishchandra S. Ramane et al., published by Sciendo

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

Keywords
Distance, transmission of a vertex, reciprocal transmission of a vertex, transmission-reciprocal transmission index