On the reformulated reciprocal sum-degree distance of graph transformations

In this paper, we study a new graph invariant named reformulated reciprocal sum-degree distance (R¯t), which is defined for a connected graph GG as R¯t(G)=∑{u,v}⊆VG(dG(u)+dG(v))1dG(u,v)+t,t≥0. On the one hand, this new graph invariant R¯t is a weight version of the tt-Harary index, i.e.,  Ht(G)=∑{u,v}⊆VG1dG(u,v)+t defined by Das et al. (2013). On the other hand, it is also the generalized version of reciprocal sum-degree distance of a connected graph, which is defined as R(G)=∑{u,v}⊆VG(dG(u)+dG(v))1dG(u,v); see Alizadeh et al. (2013) and Hua and Zhang (2012). In this paper we introduce three edge-grafting transformations to study the mathematical properties of R¯t(G). Using these nice mathematical properties, we characterize the extremal graphs among nn-vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, and vertex bipartition. Some sharp upper bounds on the reformulated reciprocal sum-degree distance of trees are determined.