Abstract
Let \(G=(V_G, E_G)\) be a simple connected graph. The reciprocal degree distance of \(G\) is defined as \(\bar{R}(G)=\sum _{\{u,v\}\subseteq V_G}(d_G(u)+d_G(v))\frac{1}{d_G(u,v)}=\sum _{u\in V_G}d_G(u)\hat{D}_G(u),\) where \(\hat{D}_G(u)=\sum _{v\in V_G\setminus \{u\}}\frac{1}{d_G(u,v)}\) is the sum of reciprocal distances from the vertex \(u.\) 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 \(G\). Using these nice mathematical properties, we characterize the extremal graphs among \(n\) 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.
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Acknowledgments
The authors would like to express their sincere gratitude to the referees for a very careful reading of this paper and for all their insightful comments, which lead a number of improvements to this paper. Financially supported by the National Natural Science Foundation of China (Grant No. 12071149) and the Special Fund for Basic Scientific Research of Central Colleges (Grant No. CCNU13F020).
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Li, S., Meng, X. Four edge-grafting theorems on the reciprocal degree distance of graphs and their applications. J Comb Optim 30, 468–488 (2015). https://doi.org/10.1007/s10878-013-9649-1
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DOI: https://doi.org/10.1007/s10878-013-9649-1