Geodesic
Completely regular clique graphs
Journal of Algebraic Combinatorics, Tome 40 (2014) no. 1, pp. 233-244.

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Let $\Gamma =(X,R)$ be a connected graph. Then $\Gamma$ is said to be a completely regular clique graph of parameters $(s,c)$ with $s\geq 1$ and $c\geq 1$, 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 this paper, we show that the parameters of $C\in\mathcal{C}$ as a completely regular code do not depend on $C\in\mathcal{C}$. As a by-product we have that all completely regular clique graphs are distance-regular whenever $\mathcal {C}$ consists of edges. We investigate the case when $\Gamma$ is distance-regular, and show that $\Gamma$ is a completely regular clique graph if and only if it is a bipartite half of a distance-semiregular graph.
Classification : 05E30, 05C12, 05C40, 05C69
Keywords: distance-regular graph, association scheme, subconstituent algebra, Terwilliger algebra, completely regular code, distance-semiregular graph 
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     author = {Suzuki, Hiroshi},
     title = {Completely regular clique graphs},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2014__40_1_a3/}
}
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Suzuki, Hiroshi. Completely regular clique graphs. Journal of Algebraic Combinatorics, Tome 40 (2014) no. 1, pp. 233-244. http://geodesic.mathdoc.fr/item/JAC_2014__40_1_a3/