Реферат: A set $C$ of vertices of a connected regular graph is called \emph{intriguing} if both $C$ and its complement $\overline C$ induce regular subgraphs. Nontrivial (when both $C$ and $\overline C$ are nonempty) intriguing sets are special cases of completely regular codes (namely, intriguing sets are completely regular codes with covering radius~$1$), introduced by Delsarte \cite{Delsarte} for distance-regular graphs and redefined by Neumaier \cite{Neumaier} in a manner applicable to an arbitrary graph. We show that in a Hamming graph over a field alphabet, additive (closed under addition) intriguing sets are equivalent to some multiset generalization of spreads, defined below. Let $S$ be a collection (multiset) of subspaces of dimension at most~$t$ of an $m$-dimensional space over~GF$(p)$. Each subspace $X$ of dimension $s$ from~$S$ is treated as the multiset of cardinality $p^t-1$ where every nonzero vector of $X$ has multiplicity $p^{t-s}$ and the zero vector has multiplicity $p^{t-s}-1$. Such~$S$ is called a \emph{$(\lambda,\mu)$-multispread} (more specifically, a $(\lambda,\mu)_p^{t,m}$-multispread) if the union of the multisets corresponding to the subspaces from~$S$ contains the zero vector with multiplicity~$\lambda$ and each nonzero vector of the space with multiplicity~$\mu$. Ordinary spreads correspond to $(0,1)$-multispreads, and $\mu$-fold spreads correspond to $(0,\mu)$-multi\-spreads~\cite[p.~83]{Hirschfeld}. An example of a $(\lambda,\mu)$-multispread with nonzero $\lambda=p^{m'-m}-1$ and $\mu=p^{m'-m}$ can be obtained from a spread of an $m'$-dimensional space, $m'>m$, by projection onto an $m$-dimensional space (we consider the projection that respects the multiplicity and preserves the cardinality of a multiset of vectors). Multispreads can be considered as a special case of vector-space partitions~\cite{El-Zanati}, and the subspaces dual to the subspaces from a multispread also form a multifold partition of the space, dual to the original multispread. The current work is devoted to the characterization of the parameters of multispreads, which is equivalent (for prime $p$) to the characterization of the parameters of additive intriguing sets in the Hamming graphs over GF$(p^t)$ and also (via duality) to the characterization of the parameters of additive one-weight codes over GF$(p^t)$. We characterize these parameters for the case $t=2$ and make a partial characterization for $t=3$ and $t=4$ (including a complete characterization for $p^t=2^3$, $3^3$, and $2^4$, where several key cases are solved computationally). This is joint work with Ivan Mogilnykh. The work is funded by the Russian Science Foundation (22-11-00266).

Библиографическая ссылка: Krotov D.
Multispreads and additive intriguing sets in Hamming graphs
The International Conference and PhD-Master Summer School "Graphs and Groups, Complexity and Convexity" 11-25 Aug 2024