Four edge-grafting theorems on the reciprocal degree distance of graphs and their applications

Let $$G=(V_G, E_G)$$ G = ( V G , E G ) be a simple connected graph. The reciprocal degree distance of $$G$$ G is defined as $$\bar{R}(G)=\sum _{\{u,v\}\subseteq V_G}(d_G(u)+d_G(v))\frac{1}{d_G(u,v)}=\sum _{u\in V_G}d_G(u)\hat{D}_G(u),$$ R ¯ ( G ) = ∑ { u , v } ⊆ V G ( d G ( u ) + d G ( v ) ) 1 d G ( u , v ) = ∑ u ∈ V G d G ( u ) D ^ G ( u ) , where $$\hat{D}_G(u)=\sum _{v\in V_G\setminus \{u\}}\frac{1}{d_G(u,v)}$$ D ^ G ( u ) = ∑ v ∈ V G \ { u } 1 d G ( u , v ) is the sum of reciprocal distances from the vertex $$u.$$ u . This novel invariant is in fact the modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. In this paper we first introduced four edge-grafting transformations to study the mathematical properties of the reciprocal degree distance of $$G$$ G . Using these nice mathematical properties, we characterize the extremal graphs among $$n$$ n vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. Some sharp upper bounds on the reciprocal degree distance of trees are determined.