Transmission-reciprocal transmission index and coindex of graphs

Abstract

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 r s ( u ) = ∑ v ∈ V ( G ) 1 d ( 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 T R T ( G ) = ∑ u v ∈ E ( G ) ( σ ( u ) r s ( u ) + σ ( v ) r s ( 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 T R T ¯ ( G ) = ∑ u v ∉ E ( G ) ( σ ( u ) r s ( u ) + σ ( v ) r s ( 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 T R T ¯ ( G ) \overline {TRT} \left( G \right) (G). Further compute this index for some standard class of graphs and obtain bounds for it.