Reciprocal Degree Distance Research Articles

Further results on the reciprocal degree distance of graphs

The reciprocal degree distance of a simple connected graph $$G=(V_G, E_G)$$G=(VG,EG) is defined as $$\bar{R}(G)=\sum _{u,v \in V_G}(\delta _G(u)+\delta _G(v))\frac{1}{d_G(u,v)}$$R¯(G)=?u,v?VG(?G(u)+?G(v))1dG(u,v), where $$\delta _G(u)$$?G(u) is the vertex degree of $$u$$u, and $$d_G(u,v)$$dG(u,v) is the distance between $$u$$u and $$v$$v in $$G$$G. The reciprocal degree distance is an additive weight version of the Harary index, which is defined as $$H(G)=\sum _{u,v \in V_G}\frac{1}{d_G(u,v)}$$H(G)=?u,v?VG1dG(u,v). In this paper, the extremal $$\bar{R}$$R¯-values on several types of important graphs are considered. The graph with the maximum $$\bar{R}$$R¯-value among all the simple connected graphs of diameter $$d$$d is determined. Among the connected bipartite graphs of order $$n$$n, the graph with a given matching number (resp. vertex connectivity) having the maximum $$\bar{R}$$R¯-value is characterized. Finally, sharp upper bounds on $$\bar{R}$$R¯-value among all simple connected outerplanar (resp. planar) graphs are determined.

Reciprocal degree distance of some graph operations

The reciprocal degree distance (RDD)‎, ‎defined for a connected graph $G$ as vertex-degree-weighted sum of the reciprocal distances‎, ‎that is‎, ‎$RDD(G) =sumlimits_{u,vin V(G)}frac{d_G(u)‎ + ‎d_G(v)}{d_G(u,v)}.$ The reciprocal degree distance is a weight version of the Harary index‎, ‎just as the degree distance is a weight version of the Wiener index‎. ‎In this paper‎, ‎we present exact formulae for the reciprocal degree distance of join‎, ‎tensor product‎, ‎strong product and wreath product of graphs in terms of other graph invariants including the degree distance‎, ‎Harary index‎, ‎the first Zagreb index and first Zagreb coindex‎. ‎Finally‎, ‎we apply some of our results to compute the reciprocal degree distance of fan graph‎, ‎wheel graph‎, ‎open fence and closed fence graphs‎.

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

Let $$G=(V_G, E_G)$$G=(VG,EG) 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}⊆VG(dG(u)+dG(v))1dG(u,v)=?u?VGdG(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?VG\{u}1dG(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.

Reciprocal Degree Distance of Grassmann Graphs

Recently, Hua et al. defined a new topological index based on degrees and inverse of distances between all pairs of vertices. They named this new graph invariant as reciprocal degree distance as 1 { , } ( ) ( ( ) ( ))[ ( , )] RDD(G) = u v V G d u  d v d u v , where the d(u,v) denotes the distance between vertices u and v. In this paper, we compute this topological index for Grassmann graphs.

On the reciprocal degree distance of graphs

In this paper, we study a new graph invariant named reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G)=∑{u,v}⊆V(G)(dG(u)+dG(v))1dG(u,v). The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. Our main purpose is to investigate extremal properties of reciprocal degree distance. We first characterize among all nontrivial connected graphs of given order the graphs with the maximum and minimum reciprocal degree distance, respectively. Then we characterize the nontrivial connected graph with given order, size and the maximum reciprocal degree distance as well as the tree, unicyclic graph and cactus with the maximum reciprocal degree distance, respectively. Finally, we establish various lower and upper bounds for the reciprocal degree distance in terms of other graph invariants including the degree distance, Harary index, the first Zagreb index, the first Zagreb coindex, pendent vertices, independence number, chromatic number and vertex-, and edge-connectivity.