1School of Mathematical Sciences, Queen Mary University of London 2School of Mathematical Sciences, Shanghai Jiao Tong University, Krasovskii Institute of Mathematics and Mechanics, Faculty of Mathematics, Chelyabinsk State University 3Faculty of Mathematics, Chelyabinsk State University
The electronic journal of combinatorics, Tome 26 (2019) no. 2
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.
Rhys J. Evans
1
;
Sergey Goryainov
2
;
Dmitry Panasenko
3
1
School of Mathematical Sciences, Queen Mary University of London
2
School of Mathematical Sciences, Shanghai Jiao Tong University,
Krasovskii Institute of Mathematics and Mechanics,
Faculty of Mathematics, Chelyabinsk State University
3
Faculty of Mathematics, Chelyabinsk State University
@article{10_37236_8189,
author = {Rhys J. Evans and Sergey Goryainov and Dmitry Panasenko},
title = {The smallest strictly {Neumaier} graph and its generalisations},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {2},
doi = {10.37236/8189},
zbl = {1414.05223},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8189/}
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TY - JOUR
AU - Rhys J. Evans
AU - Sergey Goryainov
AU - Dmitry Panasenko
TI - The smallest strictly Neumaier graph and its generalisations
JO - The electronic journal of combinatorics
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Rhys J. Evans; Sergey Goryainov; Dmitry Panasenko. The smallest strictly Neumaier graph and its generalisations. The electronic journal of combinatorics, Tome 26 (2019) no. 2. doi: 10.37236/8189
