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Alexander L. Gavrilyuk

Publications and source records attributed to Alexander L. Gavrilyuk.

At least 19 recordsLinked to original sources
arXiv2026

Intersecting Families of Spanning Trees of $K_{n,n}$

A family of spanning trees of a graph is $t$-intersecting if any pair of spanning trees in the family has $t$ or more edges in common. For sufficiently large $n$ and $t \leq n/C\log_2 n$ for some absolute constant $C>0$, we give a nearly complete characterization of the extremal $t$-intersecting families of spanning trees in balanced complete bipartite graphs with parts of order $n$. In particular, for $t=1$, we give exact bounds and a full characterization of the extremal families. For $t \geq 2$, our bounds are tight up to lower-order terms, and we show that any extremal $t$-intersecting family is of the form $\mathcal{F} \cup \mathcal{S}'$ where $\mathcal{F}$ is a family of all trees containing a fixed $t$-matching, and $\mathcal{S}'$ is a distinguished set of exceptional trees of size $|\mathcal{S}'| = o(|\mathcal{F}|)$.

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arXiv2026

Roux schemes which carry association schemes locally

A roux scheme is an association scheme formed from a special "roux" matrix and the regular permutation representation of an associated group. They were introduced by Iverson and Mixon for their connection to equiangular tight frames and doubly transitive lines. We show how roux matrices can be produced from association schemes and characterise roux schemes for which the neighbourhood of a vertex induces an association scheme possessing the same number of relations as the thin radical. An important example arises from the $64$ equiangular lines in $\mathbb{C}^8$ constructed by Hoggar which we prove is unique (determined by its parameters up to isomorphism). We also characterise roux schemes by their eigenmatrices and provide new families of roux schemes using our construction.

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arXiv2025

Extremal orthogonal arrays

It is known that a Delsarte $t$-design in a $Q$-polynomial association scheme has degree at least $\left \lceil{\frac{t}{2}}\right \rceil $. Following Ionin and Shrikhande who studied combinatorial $(2s-1)$-designs (i.e., Delsarte designs in Johnson association schemes) having exactly $s$ block intersection numbers, we call a Delsarte $(2s-1)$-design with degree $s$ extremal and study extremal orthogonal arrays, which are Delsarte designs in Hamming association schemes. It was shown by Delsarte that a $t$-design with degree $s$ and $t\geq 2s-2$ in a Hamming association scheme induces an $s$-class association scheme. We prove that an extremal orthogonal array gives rise to a fission scheme of the latter one, which has $2s-1$ or $2s$ classes. As a corollary, a new necessary condition for the existence of tight orthogonal arrays of strength $3$ is obtained. Furthermore, as a counterpart to a result of Ionin and Shrikhande, we prove an inequality for Hamming distances in extremal orthogonal arrays. The inequality is tight as shown by examples related to the Golay codes.

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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 $m\ge 6$.

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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.

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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.

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arXiv2023

Strongly regular graphs decomposable into a divisible design graph and a Hoffman coclique

In 2022, the second author found a prolific construction of strongly regular graphs, which is based on joining a coclique and a divisible design graph with certain parameters. The construction produces strongly regular graphs with the same parameters as the complement of the symplectic graph $\mathsf{Sp}(2d,q)$. In this paper, we determine the parameters of strongly regular graphs which admit a decomposition into a divisible design graph and a coclique attaining the Hoffman bound. In particular, it is shown that when the least eigenvalue of such a strongly regular graph is a prime power, its parameters coincide with those of the complement of $\mathsf{Sp}(2d,q)$. Furthermore, a generalization of the construction is discussed.

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arXiv2021

A modular equality for $m$-ovoids of elliptic quadrics

An $m$-ovoid of a finite polar space $\mathcal{P}$ is a set $\mathcal{O}$ of points such that every maximal subspace of $\mathcal{P}$ contains exactly $m$ points of $\mathcal{O}$. In the case when $\mathcal{P}$ is an elliptic quadric $\mathcal{Q}^-(2r+1, q)$ of rank $r$ in $\mathbb{F}_q^{2r+2}$, we prove that an $m$-ovoid exists only if $m$ satisfies a certain modular equality, which depends on $q$ and $r$. This condition rules out many of the possible values of $m$. Previously, only a lower bound on $m$ was known, which we slightly improve as a byproduct of our method. We also obtain a characterization of the $m$-ovoids of $\mathcal{Q}^{-}(7,q)$ for $q = 2$ and $(m, q) = (4, 3)$.

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arXiv2021

Signed analogue of line graphs and their smallest eigenvalues

In this paper, we show that every connected signed graph with smallest eigenvalue strictly greater than $-2$ and large enough minimum degree is switching equivalent to a complete graph. This is a signed analogue of a theorem of Hoffman. The proof is based on what we call Hoffman's limit theorem which we formulate for Hermitian matrices, and also the extension of the concept of Hoffman graph and line graph for the setting of signed graphs.

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arXiv2020

On the multiplicities of digraph eigenvalues

We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first shown by Delsarte, Goethals, and Seidel (1977) and extended by Bell and Rowlinson (2003). In doing so, we also adapt the Blokhuis' theory (1983) of harmonic analysis in real hyperbolic spaces to that in complex hyperbolic spaces.

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arXiv2020

The Weisfeiler-Leman dimension of distance-hereditary graphs

A graph is said to be distance-hereditary if the distance function in every connected induced subgraph is the same as in the graph itself. We prove that the ordinary Weisfeiler-Leman algorithm correctly tests the isomorphism of any two graphs if one of them is distance-hereditary; more precisely, the Weisfeiler-Leman dimension of the class of finite distance-hereditary graphs is equal to $2$. The previously best known upper bound for the dimension was $7$.

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arXiv2019

On tight sets of hyperbolic quadrics

We prove that the parameter $x$ of a tight set $\mathcal{T}$ of a hyperbolic quadric $\mathsf{Q}^+(2n+1,q)$ of an odd rank $n+1$ satisfies ${x\choose 2}+w(w-x)\equiv 0\mod q+1$, where $w$ is the number of points of $\mathcal{T}$ in any generator of $\mathsf{Q}^+(2n+1,q)$. As this modular equation should have an integer solution in $w$ if such a $\mathcal{T}$ exists, this condition rules out roughly at least one half of all possible parameters $x$. It generalizes a previous result by the author and K. Metsch shown for tight sets of a hyperbolic quadric $\mathsf{Q}^+(5,q)$ (also known as Cameron-Liebler line classes in $\mathrm{PG}(3,q)$).

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