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Further results on the reciprocal degree distance of graphs

Author

Listed:
  • Shuchao Li

    (Central China Normal University)

  • Huihui Zhang

    (Central China Normal University)

  • Minjie Zhang

    (Hubei Institute of Technology)

Abstract

The reciprocal degree distance of a simple connected graph $$G=(V_G, E_G)$$ G = ( V G , E G ) 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 ∈ V G ( δ G ( u ) + δ G ( v ) ) 1 d G ( u , v ) , where $$\delta _G(u)$$ δ G ( u ) is the vertex degree of $$u$$ u , and $$d_G(u,v)$$ d G ( 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 ∈ V G 1 d G ( 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.

Suggested Citation

  • Shuchao Li & Huihui Zhang & Minjie Zhang, 2016. "Further results on the reciprocal degree distance of graphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 648-668, February.
  • Handle: RePEc:spr:jcomop:v:31:y:2016:i:2:d:10.1007_s10878-014-9780-7
    DOI: 10.1007/s10878-014-9780-7
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    References listed on IDEAS

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    1. Ailin Hou & Shuchao Li & Lanzhen Song & Bing Wei, 2011. "Sharp bounds for Zagreb indices of maximal outerplanar graphs," Journal of Combinatorial Optimization, Springer, vol. 22(2), pages 252-269, August.
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