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Completely regular clique graphs, II

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Abstract

Let \(\varGamma = (X,R)\) be a connected graph. Then \(\varGamma \) is said to be a completely regular clique graph of parameters (s, c) with \(s\ge 1\) and \(c\ge 1\), if there is a collection \({\mathcal {C}}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \({\mathcal {C}}\). In the previous paper (Suzuki in J Algebr Combin 40:233–244, 2014), we showed, among other things, that a completely regular clique graph is distance-regular if and only if it is a bipartite half of a certain distance-semiregular graph. In this paper, we show that a completely regular clique graph with respect to \({\mathcal {C}}\) is distance-regular if and only if every \({\mathcal {T}}(C)\)-module of endpoint zero is thin for all \(C\in {\mathcal {C}}\). We also discuss the relation between a \({\mathcal {T}}(C)\)-module of endpoint 0 and a \({\mathcal {T}}(x)\)-module of endpoint 1 and study examples of completely regular clique graphs.

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Notes

  1. It will be shown that each member of \({\mathcal {C}}\) is a maximal clique if \(d(\varGamma ) >1\). See a remark preceding Lemma 5.

  2. This condition is not stated in [12]. If we do not have this condition, we need to allow \({\mathcal {C}}\) to be a multi-set. We do not need this condition if \(c_2^Y\) exists and \(d^Y\ge 3\).

  3. This class can be seen as a special case of C.

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Acknowledgments

This research was partially supported by the Grant-in-Aid for Scientific Research (No. 21540023), Japan Society of the Promotion of Science.

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  1. International Christian University, Mitaka, Tokyo, 181-8585, Japan

    Hiroshi Suzuki

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  1. Hiroshi Suzuki
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Correspondence to Hiroshi Suzuki.

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Suzuki, H. Completely regular clique graphs, II. J Algebr Comb 43, 417–445 (2016). https://doi.org/10.1007/s10801-015-0639-5

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