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 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 a...
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| Format: | Article |
| Language: | English |
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Sciendo
2022-08-01
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| Series: | Acta Universitatis Sapientiae: Informatica |
| Subjects: | |
| Online Access: | https://doi.org/10.2478/ausi-2022-0006 |
| _version_ | 1811185964709576704 |
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| author | Ramane Harishchandra S. Kitturmath Deepa V. Bhajantri Kavita |
| author_facet | Ramane Harishchandra S. Kitturmath Deepa V. Bhajantri Kavita |
| author_sort | Ramane Harishchandra S. |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T13:39:27Z |
| format | Article |
| id | doaj.art-12fa419d51494f9ab42e1f55b7a31603 |
| institution | Directory Open Access Journal |
| issn | 2066-7760 |
| language | English |
| last_indexed | 2024-04-11T13:39:27Z |
| publishDate | 2022-08-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Acta Universitatis Sapientiae: Informatica |
| spelling | doaj.art-12fa419d51494f9ab42e1f55b7a316032022-12-22T04:21:21ZengSciendoActa Universitatis Sapientiae: Informatica2066-77602022-08-011418410310.2478/ausi-2022-0006Transmission-reciprocal transmission index and coindex of graphsRamane Harishchandra S.0Kitturmath Deepa V.1Bhajantri Kavita2Department of Mathematics, Karnatak University, Dharwad-580003, IndiaDepartment of Mathematics, Karnatak University, Dharwad-580003, IndiaDepartment of Mathematics, JSS Banashankari Arts, Commerce and Shantikumar Gubbi Science College, Vidyagiri, Dharwad-580004, IndiaThe 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.https://doi.org/10.2478/ausi-2022-0006distancetransmission of a vertexreciprocal transmission of a vertextransmission-reciprocal transmission index05c1205c09 |
| spellingShingle | Ramane Harishchandra S. Kitturmath Deepa V. Bhajantri Kavita Transmission-reciprocal transmission index and coindex of graphs Acta Universitatis Sapientiae: Informatica distance transmission of a vertex reciprocal transmission of a vertex transmission-reciprocal transmission index 05c12 05c09 |
| title | Transmission-reciprocal transmission index and coindex of graphs |
| title_full | Transmission-reciprocal transmission index and coindex of graphs |
| title_fullStr | Transmission-reciprocal transmission index and coindex of graphs |
| title_full_unstemmed | Transmission-reciprocal transmission index and coindex of graphs |
| title_short | Transmission-reciprocal transmission index and coindex of graphs |
| title_sort | transmission reciprocal transmission index and coindex of graphs |
| topic | distance transmission of a vertex reciprocal transmission of a vertex transmission-reciprocal transmission index 05c12 05c09 |
| url | https://doi.org/10.2478/ausi-2022-0006 |
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