

2022ary quasigroups and related topics
April 6, 2018, Novosibirsk, Sobolev Institute of Mathematics, room 135






Perfect 2colorings of Hamming graphs with eigenvalue $\lambda_2$
A. A. Valyuzhenich^{}, I. Yu. Mogil'nykh^{} 
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Abstract:
An eigenvalue of the perfect coloring is an eigenvalue of its parameter matrix. Perfect 2colorings of Hamming graphs with eigenvalue $\lambda_1$ were classified earlier in [1].
In this paper we prove that every perfect 2coloring of the graph $H(n, q)$ with eigenvalue $\lambda_2$ are reduced to perfect 2colorings $H(3,q)$ by removing nonessential directions, except for colorings constructed from perfect 2colorings of $H (2, q)$ by means of substitution swatches and colorings obtained from partitions of $H (4,2)$ into two cycles. A classification of perfect 2colorings $H(n,q)$ with an eigenvalue $\lambda_2$ for $q = 2,3,4$ is found.
[1] A. D. Meyerowitz, Cyclebalance partitions for distanceregular graphs, Discrete Math 264:13 (2003), 149–165. https://doi.org/10.1016/S0012365X(02)005575

