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Grigory Ryabov

Publications and source records attributed to Grigory Ryabov.

At least 37 records · Page 2Linked to original sources
arXiv2021

On WL-rank and WL-dimension of some Deza dihedrants

The WL-rank of a graph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. The WL-dimension of $Γ$ is defined to be the smallest positive integer $m$ for which $Γ$ is identified by the $m$-dimensional Weisfeiler-Leman algorithm. We establish that some families of strictly Deza dihedrants have WL-rank $4$ or $5$ and WL-dimension $2$. Computer calculations imply that every strictly Deza dihedrant with at most $59$ vertices is circulant or belongs to one of the above families. We also construct a new infinite family of strictly Deza dihedrants whose WL-rank is a linear function of the number of vertices.

math.CO↗
arXiv2021

On pseudofrobenius imprimitive association schemes

An (association) scheme is said to be Frobenius if it is the scheme of a Frobenius group. A scheme which has the same tensor of intersection numbers as some Frobenius scheme is said to be pseudofrobenius. We establish a necessary and sufficient condition for an imprimitive pseudofrobenius scheme to be Frobenius. We also prove strong necessary conditions for existence of an imprimitive pseudofrobenius scheme which is not Frobenius. As a byproduct, we obtain a sufficient condition for an imprimitive Frobenius group $G$ with abelian kernel to be determined up to isomorphism only by the character table of $G$. Finally, we prove that the Weisfeiler-Leman dimension of a circulant graph with $n$ vertices and Frobenius automorphism group is equal to $2$ unless $n\in \{p,p^2,p^3,pq,p^2q\}$, where $p$ and $q$ are distinct primes.

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arXiv2021

On WL-rank of Deza Cayley graphs

The WL-rank of a digraph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. We construct a new infinite family of strictly Deza Cayley graphs for which the WL-rank is equal to the number of vertices. The graphs from this family are divisible design and integral.

math.CO↗
arXiv2021

Normal Cayley digraphs of cyclic groups with CI-property

A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $T\subseteq G$, $Cay(G,S)\cong Cay(G,T)$ implies that there is $σ\in Aut(G)$ such that $S^σ=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if $8\nmid n$, and is a NCI-group if and only if either $n=8$ or $8\nmid n$.

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arXiv2020

On WL-rank and WL-dimension of some Deza circulant graphs

The WL-rank of a digraph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. The WL-dimension of $Γ$ is defined to be the smallest positive integer $m$ for which $Γ$ is identified by the $m$-dimensional Weisfeiler-Leman algorithm. We classify the Deza circulant graphs of WL-rank $4$. In additional, it is proved that each of these graphs has WL-dimension at most $3$. Finally, we establish that some families of Deza circulant graphs have WL-rank $5$ or $6$ and WL-dimension at most $3$.

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arXiv2020

The Cayley isomorphism property for the group $C^5_2\times C_p$

A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_2^5\times C_p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$. Together with the previously obtained results, this implies that a group $G$ of order $32p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$ and $G\cong C_2^5\times C_p$.

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arXiv2020

The Cayley isomorphism property for the group $C_4\times C_p^2$

A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_4\times C_p^2$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$.

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arXiv2019

The group $C^4_p\times C_q$ is a DCI-group

We prove that the group $C_p^4\times C_q$ is a DCI-group for distinct primes $p$ and $q$, that is, two Cayley digraphs over $C_p^4 \times C_q$ are isomorphic if and only if their connection sets are conjugate by a group automorphism.

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arXiv2019

Separability of Schur rings over abelian groups of odd order

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. A finite group $G$ is said to be separable with respect to $\mathcal{K}$ if every $S$-ring over $G$ is separable with respect to $\mathcal{K}$. We prove that every abelian group $G$ of order $9p$, where $p$ is a prime, is separable with respect to the class of all finite abelian groups. Modulo previously obtained results, this completes a classification of noncyclic abelian groups of odd order that are separable with respect to the class of all finite abelian groups. Also this implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.

math.CO↗
arXiv2019

On Cayley representations of finite graphs over abelian p-groups

We construct a polynomial-time algorithm which given a graph $Γ$ finds the full set of non-equivalent Cayley representations of $Γ$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies that the recognition and the isomorphism problems for Cayley graphs over $D$ can be solved in polynomial time.

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arXiv2019

On separable Schur rings over abelian groups

A finite group is said to be weakly separable if every algebraic isomorphism between two $S$-rings over this group is induced by a combinatorial isomorphism. In the paper we prove that every abelian weakly separable group belongs to one of several explicitly given families only.

math.GR↗
arXiv2019

On separable abelian $p$-groups

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. A finite group is said to be \emph{separable} with respect to $\mathcal{K}$ if every $S$-ring over this group is separable with respect to $\mathcal{K}$. We provide a complete classification of abelian $p$-groups separable with respect to the class of abelian groups.

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arXiv2018

Separability of Schur rings over an abelian group of order 4p

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every its algebraic isomorphism to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. We prove that every Schur ring over an abelian group $G$ of order $4p$, where $p$ is a prime, is separable with respect to the class of abelian groups. This implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.

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arXiv2018

$CI$-property for decomposable Schur rings over an abelian group

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a $CI$-Schur ring. By using this condition we reprove in a short way known results on the $CI$-property for decomposable Schur rings over an elementary abelian group of rank at most $5$.

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arXiv2017

Abelian Schur groups of odd order

A finite group $G$ is called a Schur group if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. It is proved that the group $C_3\times C_3\times C_p$ is Schur for any prime $p$. Together with earlier results, this completes a classification of the abelian Schur groups of odd order.

math.GR↗
arXiv2017

On separability of Schur rings over abelian p-groups

An $S$-ring (Schur ring) is called separable with respect to a class of $S$-rings $\mathcal{K}$ if it is determined up to isomorphism in $\mathcal{K}$ only by the tensor of its structure constants. An abelian group is said to be separable if every $S$-ring over this group is separable with respect to the class of $S$-rings over abelian groups. Let $C_n$ be a cyclic group of order $n$ and $G$ be a noncylic abelian $p$-group. From the previously obtained results it follows that if $G$ is separable then $G$ is isomorphic to $C_p\times C_{p^k}$ or $C_p\times C_p\times C_{p^k}$, where $p\in \{2,3\}$ and $k\geq 1$. We prove that the groups $D=C_p\times C_{p^k}$ are separable whenever $p\in \{2,3\}$. From this statement we deduce that a given Cayley graph over $D$ and a given Cayley graph over an arbitrary abelian group one can check whether these graphs are isomorphic in time $|D|^{O(1)}$.

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