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The Power Index of a Graph

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Abstract

The power index \(\Theta (\Gamma )\) of a graph \(\Gamma \) is the least order of a group G such that \(\Gamma \) can embed into the power graph of G. Furthermore, this group G is \(\Gamma \) -optimal if G has order \(\Theta (\Gamma )\). We say that \(\Gamma \) is power-critical if its order is equal to \(\Theta (\Gamma )\). This paper focuses on the power indices of complete graphs, complete bipartite graphs and 1-factors. We classify all power-critical graphs \(\Gamma '\) in these three families, and give a necessary and sufficient condition for \(\Gamma '\)-optimal groups.

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Acknowledgements

We are grateful to the referees for many useful suggestions and comments. We thank LetPub (http://www.letpub.com) for its linguistic assistance during the revision of this manuscript. This work was carried out during Ma’s visit to the Beijing Normal University (Dec. 2016–Jan. 2017). Ma’s research was supported by National Natural Science Foundation of China (61472471) and Innovation Talent Promotion Plan of Shaanxi Province for Young Sci-Tech New Star (No. 2017KJXX-60). Wang’s research was supported by National Natural Science Foundation of China (11371204, 11671043) and the Fundamental Research Funds for the Central University of China. Feng’s research was supported by National Natural Science Foundation of China (11701281).

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Authors and Affiliations

  1. School of Science, Xi’an Shiyou University, Xi’an, 710065, China

    Xuanlong Ma

  2. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China

    Min Feng

  3. Sch. Math. Sci. and Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China

    Kaishun Wang

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  1. Xuanlong Ma
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  2. Min Feng
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  3. Kaishun Wang
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Correspondence to Min Feng.

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Ma, X., Feng, M. & Wang, K. The Power Index of a Graph. Graphs and Combinatorics 33, 1381–1391 (2017). https://doi.org/10.1007/s00373-017-1851-y

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  • DOI: https://doi.org/10.1007/s00373-017-1851-y

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