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On the Degree Distance of Unicyclic Graphs with Given Matching Number

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Abstract

Let \(G=(V_G, E_G)\) be a connected graph. The degree distance of \(G\) is defined as \(D'(G)= \sum _{\{u,v\} \subseteq V_G}(d_G(v)+d_G(v))d_G(u,v)\), where \(d_G(v)\) is the degree of vertex \(v, d_G(u, v)\) denotes the distance between \(u\) and \(v\) in \(G\). This parameter was introduced, independently, by Dobrynin and Kochetova and by Gutman as a weighted version of the Wiener index. Feng et al. (Graphs Comb 29(3):449–462, 2013) characterized \(n\)-vertex unicyclic graphs with given matching number having the minimal degree distance. As a continuance of it, in this paper the \(n\)-vertex unicyclic graphs of given matching number with the second and third minimal degree distance are identified respectively.

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Acknowledgments

The authors would like to express their sincere gratitude to both of the referees for a very careful reading of this paper and for all their insightful comments, which led to number of improvements to this paper. This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 11271149,11371162), the Program for New Century Excellent Talents in University (Grant No. NCET-13-0817) and the Special Fund for Basic Scientific Research of Central Colleges (Grant No. CCNU13F020).

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  1. Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Shuchao Li, Yibing Song & Huihui Zhang

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  1. Shuchao Li
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  2. Yibing Song
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  3. Huihui Zhang
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Correspondence to Shuchao Li.

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Li, S., Song, Y. & Zhang, H. On the Degree Distance of Unicyclic Graphs with Given Matching Number. Graphs and Combinatorics 31, 2261–2274 (2015). https://doi.org/10.1007/s00373-015-1527-4

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  • DOI: https://doi.org/10.1007/s00373-015-1527-4

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