Another construction of edge-regular graphs with regular cliques

Gary R.W. Greaves; Jack H. Koolen

doi:10.1016/j.disc.2018.09.032

Another construction of edge-regular graphs with regular cliques

Discrete Mathematics ◽

10.1016/j.disc.2018.09.032 ◽

2019 ◽

Vol 342 (10) ◽

pp. 2818-2820

Author(s):

Gary R.W. Greaves ◽

Jack H. Koolen

Keyword(s):

Regular Graphs ◽

Regular Cliques

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Edge-regular graphs with regular cliques

European Journal of Combinatorics ◽

10.1016/j.ejc.2018.04.004 ◽

2018 ◽

Vol 71 ◽

pp. 194-201 ◽

Cited By ~ 1

Author(s):

Gary R.W. Greaves ◽

Jack H. Koolen

Keyword(s):

Regular Graphs ◽

Regular Cliques

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The Smallest Strictly Neumaier Graph and its Generalisations

The Electronic Journal of Combinatorics ◽

10.37236/8189 ◽

2019 ◽

Vol 26 (2) ◽

Author(s):

Rhys J. Evans ◽

Sergey Goryainov ◽

Dmitry Panasenko

Keyword(s):

Positive Integer ◽

Regular Graph ◽

Regular Graphs ◽

Current Interest ◽

Polar Graph ◽

Strongly Regular ◽

Regular Cliques

A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by A. Neumaier in the 1980s and attracting current interest. We thus define a Neumaier graph to be an non-complete edge-regular graph containing a regular clique, and a strictly Neumaier graph to be a non-strongly regular Neumaier graph. We first prove some general results on Neumaier graphs and their feasible parameter tuples. We then apply these results to determine the smallest strictly Neumaier graph, which has $16$ vertices. Next we find the parameter tuples for all strictly Neumaier graphs having at most $24$ vertices. Finally, we give two sequences of graphs, each with $i^{th}$ element a strictly Neumaier graph containing a $2^{i}$-regular clique (where $i$ is a positive integer) and having parameters of an affine polar graph as an edge-regular graph. This answers questions recently posed by G. Greaves and J. Koolen.

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