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Ilia Ponomarenko

Publications and source records attributed to Ilia Ponomarenko.

At least 19 recordsLinked to original sources
arXiv2026

On circulant ternary coherent configurations of prime degree

Ternary coherent configurations are, on the one hand, a special case of multidimensional coherent configurations introduced by L. Babai (2016), and, on the other hand, a natural generalization of association schemes on triples introduced by D. M. Mesner and P. Bhattacharya (1990). A ternary coherent configuration X is said to be circulant if the automorphism group Aut(X) of X has a regular cyclic subgroup, and schurian if the classes of X are the orbits of the componentwise action of the group Aut(X) on triples of points of X. It is proved that any circulant ternary coherent configuration X of prime degree p is schurian with the possible exception of the case when X is an association schemes on triples and either Aut(X) = AGL1(p) and p = +1, or -1 (mod 8), or Aut(X) is a proper subgrou of AGL1(p).

math.CO↗
arXiv2025

The automorphism groups and identification of some Generalized Paley Graphs

The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{AΓL}}(1,q)$ whenever $q$ is sufficiently large relative to $k$. Furthermore, under the same conditions, the Weisfeiler-Leman dimension of these graphs is proved to be at most $5$. In particular, the same bound holds for the Van Lint-Schrijver graphs.

math.CO↗
arXiv2024

Notes on $B$-groups

Following Wielandt, a finite group $G$ is called a $B$-group (Burnside group) if every primitive group containing a regular subgroup isomorphic to $G$ is doubly transitive. Using a method of Schur rings, Wielandt proved that every abelian group of composite order which has at least one cyclic Sylow subgroup is a $B$-group. Since then, other infinite families of $B$-groups were found by the same method. A simple analysis of the proofs of these results shows that in all of them a stronger statement was proved for the group $G$ under consideration: every primitive Schur ring over $G$ is trivial. A finite group $G$ possessing the latter property, we call $BS$-group (Burnside-Schur group). In the present note, we give infinitely many examples of $B$-groups which are not $BS$-groups.

math.GR↗
arXiv2024

Cartesian products of graphs and their coherent configurations

The coherent configuration $\mathsf{WL}(X)$ of a graph $X$ is the smallest coherent configuration on the vertices of $X$ that contains the edge set of $X$ as a relation. The aim of the paper is to study $\mathsf{WL}(X)$ when $X$ is a Cartesian product of graphs. The example of a Hamming graph shows that, in general, $\mathsf{WL}(X)$ does not coincide with the tensor product of the coherent configurations of the factors. We prove that if $X$ is ``closed'' with respect to the $6$-dimensional Weisfeiler-Leman algorithm, then $\mathsf{WL}(X)$ is the tensor product of the coherent configurations of certain graphs related to the prime decomposition of $X$. This condition is trivially satisfied for almost all graphs. In addition, we prove that the property of a graph ``to be decomposable into a Cartesian product of $k$ connected prime graphs'' for some $k\ge 1$ is recognized by the $m$-dimensional Weisfeiler-Leman algorithm for all $m\ge 6$.

math.CO↗
arXiv2024

On multivalued groups of order 3

A complete classification of the multivalued coset groups of order $3$ is given. The proof is based on the classification of rank $3$ groups having regular normal subgroups.

math.GR↗
arXiv2024

On the Weisfeiler-Leman dimension of circulant graphs

A circulant graph is a Cayley graph of a finite cyclic group. The Weisfeiler-Leman-dimension of a circulant graph $X$ with respect to the class of all circulant graphs is the smallest positive integer~$m$ such that the $m$-dimensional Weisfeiler-Leman algorithm correctly tests the isomorphism between $X$ and any other circulant graph. It is proved that for a circulant graph of order $n$ this dimension is less than or equal to $Ω(n)+3$, where $Ω(n)$ is the number of prime divisors of~$n$.

math.CO↗
arXiv2024

A linear programming bound for sum-rank metric codes

We derive a linear programming bound on the maximum cardinality of error-correcting codes in the sum-rank metric. Based on computational experiments on relatively small instances, we observe that the obtained bounds outperform all previously known bounds.

math.CO↗
arXiv2024

Closures of permutation groups with restricted nonabelian composition factors

Given a permutation group $G$ on a finite set $Ω$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $Ω$ having the same orbits in the induced action on $Ω^k$ as $G$. Recall that a group is $\mathrm{Alt}(d)$-free if it does not contain a section isomorphic to the alternating group of degree $d$. Motivated by some problems in computational group theory, we prove that the $k$-closure of an $\mathrm{Alt}(d)$-free group is again $\mathrm{Alt}(d)$-free for $k \geq 4$ and $d \geq 25$.

math.GR↗
arXiv2023

A large family of strongly regular graphs with small Weisfeiler-Leman dimension

In 2002, D. Fon-Der-Flaass constructed a prolific family of strongly regular graphs. In this paper, we prove that for infinitely many natural numbers $n$, this family contains $n^{Ω(n^{2/3})}$ strongly regular $n$-vertex graphs $X$ with the same parameters, which satisfy the following condition: an isomorphism between $X$ and any other graph can be verified by the $4$-dimensional Weisfeiler-Leman algorithm.

math.CO↗
arXiv2023

On the Weisfeiler algorithm of depth-$1$ stabilization

An origin of the multidimensional Weisfeiler-Leman algorithm goes back to a refinement procedure of deep stabilization, introduced by B. Weisfeiler in a paper included in the collective monograph ``On construction and identification of graphs"(1976). This procedure is recursive and the recursion starts from an algorithm of depth-$1$ stabilization, which has never been discussed in the literature. A goal of the present paper is to show that a simplified algorithm of the depth-$1$ stabilization has the same power as the $3$-dimensional Weisfeiler-Leman algorithm. It is proved that the class of coherent configurations obtained at the output of this simplified algorithm coincides with the class introduced earlier by the third author. As an application we also prove that if there exist at least two nonisomorphic projective planes of order $q$, then the Weisfeiler-Leman dimension of the incidence graph of any projective plane of order $q$ is at least $4$.

math.CO↗
arXiv2023

On the Weisfeiler-Leman dimension of some polyhedral graphs

Let $m$ be a positive integer, $X$ a graph with vertex set $Ω$, and ${\rm WL}_m(X)$ the coloring of the Cartesian $m$-power $Ω^m$, obtained by the $m$-dimensional Weisfeiler-Leman algorithm. The ${\rm WL}$-dimension of the graph $X$ is defined to be the smallest $m$ for which the coloring ${\rm WL}_m(X)$ determines $X$ up to isomorphism. It is known that the ${\rm WL}$-dimension of any planar graph is $2$ or $3$, but no planar graph of ${\rm WL}$-dimension $3$ is known. We prove that the ${\rm WL}$-dimension of a polyhedral (i.e., $3$-connected planar) graph $X$ is at most $2$ if the color classes of the coloring ${\rm WL}_2(X)$ are the orbits of the componentwise action of the group ${\rm Aut}(X)$ on $Ω^2$.

math.CO↗
arXiv2023

On multidimensional Schur rings of finite groups

For any finite group $G$ and a positive integer $m$, we define andstudy a Schur ring over the direct power $G^m$, which gives an algebraic interpretation of the partition of $G^m$ obtained by the $m$-dimensional Weisfeiler-Leman algorithm. It is proved that this ring determines the group $G$ up to isomorphism if $m\ge 3$, and approaches the Schur ring associated with the group $Aut(G)$ acting on $G^m$ naturally if $m$ increases. It turns out that the problem of finding this limit ring is polynomial-time equivalent to the group isomorphism problem.

math.GR↗
arXiv2022

On the WL-dimension of circulant graphs of prime power order

The WL-dimension of a graph X is the smallest positive integer m such that the m-dimensional Weisfeiler-Leman algorithm correctly tests the isomorphism between X and any other graph. It is proved that the WL-dimension of any circulant graph of prime power order is at most 3, and this bound cannot be reduced. The proof is based on using theories of coherent configurations and Cayley schemes over a cyclic group.

math.CO↗
arXiv2022

Testing isomorphism of chordal graphs of bounded leafage is fixed-parameter tractable

The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter. In the process we introduce the problem of isomorphism testing for higher-order hypergraphs and show that finding the automorphism group of order-$k$ hypergraphs with vertex color classes of size $b$ is fixed parameter tractable for any constant $k$ and $b$ as fixed parameter.

cs.DS↗