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Andrea Švob

Publications and source records attributed to Andrea Švob.

17 recordsLinked to original sources
arXiv2026

Switching graphs and designs

Switching methods can be seen as certain local transformations that do not alter their basic parameters of a combinatorial structure. Efforts have been devoted in the literature to relate and unify the switching theories for codes and designs, and also for Hadamard matrices and graphs. The combinatorial structures we consider in this paper are graphs and designs. We show an extension of known switching method for constructing 2-designs to divisible designs, and then provide some examples of its application. Moreover, we prove several equivalences between switching methods for graphs and designs, and as a byproduct, we obtain a new switching method to obtain 2-designs.

math.CO↗
arXiv2026

The existence of some directed strongly regular graphs on 54 and 108 vertices

In this paper, we prove the existence of directed strongly regular graphs with parameters (108,11,3,2,1), (108,14,10,0,2), (108,22,12,6,4), (108,23,9,8,4), (108,25,15,8,5), (108,34,18,12,10), (108,38,22,12,14), (108,39,23,14,14), (108,41,35,16,15), (108,42,33,18,15) and (108,46,22,19,20). Further, we obtain directed strongly regular graphs with parameters (54,10,4,1,2), (54,11,4,3,2) and (54,16,7,4,5). The constructions are obtained by considering finite groups acting transitively on 54 and 108 vertices.

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arXiv2026

Complex generalised weighing matrices in centraliser algebras of monomial representations

An $n \times n$ matrix $W$ with exactly $w$ non-zero entries taken from the set of $k^{\rm th}$ complex roots of unity in each row and column satisfying $WW^{\ast} = wI_n$ is a complex generalised weighing matrix $CGW(n,w;k)$. We study such matrices through the centraliser algebras of monomial representations of finite groups. Using an exhaustive search over the linear characters of Schur covers, we classify, up to monomial equivalence, the complex generalised weighing matrices admitting a primitive group of rank at most five and degree at most $80$ acting by strong automorphisms, for coefficient orders $k \leq 6$, with partial results for larger degrees $100$. The census recovers known infinite families related to projective and affine finite geometries, describes infinite families related to Hamming schemes and settles the existence of some small open cases enumerated in the literature. We construct quantum error-correcting codes from the these matrices and determine their minimum distances exactly in all cases.

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arXiv2026

A universal theory of switching for combinatorial objects, and applications to complex Hadamard matrices

The concept of switching has arisen in several different areas within combinatorics. The act of switching usually transforms a combinatorial object into a non-isomorphic object of the same type, in a way that some key property is preserved. Godsil-McKay switching of graphs preserves the spectrum, switching of designs preserves their parameters, and switching of binary codes preserves the minimum distance. For Hadamard matrices, the switching techniques introduced by Orrick proved to be an incredibly powerful tool in generating inequivalent Hadamard matrices. In this paper, we introduce a universal definition of switching that can be adapted to incorporate these known types of switching. Through this language, we extend Orrick's methods to Butson Hadamard and complex Hadamard matrices. We introduce switchings of these matrices that can be used to construct new, inequivalent matrices. We also consider the concept of trades in complex Hadamard matrices in this terminology, and address an open problem on the permissible size of a trade.

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arXiv2024

Construction of directed strongly regular graphs via their orbit matrices and genetic algorithm

In this paper, we introduce orbit matrices of directed strongly regular graphs (DSRGs). Further, we propose a method of constructing directed strongly regular graphs with prescribed automorphism group using genetic algorithm. In the construction, we use orbit matrices, i.e. quotient matrices related to equitable partitions of adjacency matrices of putative directed strongly regular graphs induced by an action of a prescribed automorphism group. Further, we apply this method to construct directed strongly regular graphs with parameters $(36,10,5,2,3)$, $(52,12,3,2,3)$, $(52,15,6,5,6)$, $(55,20,8,6,8)$ and $(55,24,12,11,10)$.

math.CO↗
arXiv2024

Constructions of self-orthogonal and LCD subspace codes

Recently, the notions of self-orthogonal subspace codes and LCD subspace codes were introduced, and LCD subspace codes obtained from mutually unbiased weighing matrices were studied. In this paper, we provide a method of constructing self-orthogonal and LCD subspace codes from a set of matrices under certain conditions. In particular, we give constructions of self-orthogonal and LCD subspace codes from mutually quasi-unbiased weighing matrices, linked systems of symmetric designs, and linked systems of symmetric group divisible designs, Deza graphs and their equitable partitions.

math.CO↗
arXiv2023

Doubly even self-orthogonal codes from quasi-symmetric designs

In this paper, we give a construction of doubly even self-orthogonal codes from quasi-symmetric designs. Further, we study orbit matrices of quasi-symmetric designs and give a construction of doubly even self-orthogonal codes from orbit matrices of quasi-symmetric designs of Blokhuis-Haemers type.

math.CO↗
arXiv2023

Neighbour-transitive codes in Kneser graphs

A code $C$ is a subset of the vertex set of a graph and $C$ is $s$-neighbour-transitive if its automorphism group ${\rm Aut}(C)$ acts transitively on each of the first $s+1$ parts $C_0,C_1,\ldots,C_s$ of the distance partition $\{C=C_0,C_1,\ldots,C_ρ\}$, where $ρ$ is the covering radius of $C$. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let $Ω$ be the underlying set on which the Kneser graph $K(n,k)$ is defined. Our first main result says that if $C$ is a $2$-neighbour-transitive code in $K(n,k)$ such that $C$ has minimum distance at least $5$, then $n=2k+1$ (i.e., $C$ is a code in an odd graph) and $C$ lies in a particular infinite family or is one particular sporadic example. We then prove several results when $C$ is a neighbour-transitive code in the Kneser graph $K(n,k)$. First, if ${\rm Aut}(C)$ acts intransitively on $Ω$ we characterise $C$ in terms of certain parameters. We then assume that ${\rm Aut}(C)$ acts transitively on $Ω$, first proving that if $C$ has minimum distance at least $3$ then either $K(n,k)$ is an odd graph or ${\rm Aut}(C)$ has a $2$-homogeneous (and hence primitive) action on $Ω$. We then assume that $C$ is a code in an odd graph and ${\rm Aut}(C)$ acts imprimitively on $Ω$ and characterise $C$ in terms of certain parameters. We give examples in each of these cases and pose several open problems.

math.CO↗
arXiv2022

$q$-Analogs of strongly regular graphs

We introduce the notion of q-analogs of strongly regular graphs and give several examples of such structures. We prove a necessary condition on the parameters, show the connection to designs over finite fields, and present a classification.

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arXiv2022

Switching for 2-designs

In this paper we introduce a switching for 2-designs. We illustrate this method by applying it to some symmetric (64,28,12) designs. In that way we obtain six new symmetric (64,28,12) designs. Further, we show that this type of switching can be applied to any symmetric design related to a Bush-type Hadamard matrix.

math.CO↗
arXiv2021

New constructions of divisible design Cayley graphs

Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. Further, divisible design graphs which can be obtained as Cayley graphs were recently studied by Kabanov and Shalaginov. In this paper we give new constructions of divisible design Cayley graphs and classify divisible design Cayley graphs on $v \le 27$ vertices.

math.CO↗
arXiv2021

New symmetric 2-(176,50,14) designs

In this paper we construct two new symmetric designs with parameters 2-(176,50,14) as designs invariant under certain subgroups of the full automorphism group of the Higman design. One is self-dual and has the full automorphism group of size 11520 and other is not self-dual and has the full automorphism group of size 2520.

math.CO↗
arXiv2020

Strongly regular graphs with parameters (81,30,9,12) and a new partial geometry pg(5,5,2)

Twelve new strongly regular graphs with parameters (81,30,9,12) are found as graphs invariant under certain subgroups of the automorphism groups of the two previously known graphs that arise from 2-weight codes. One of these new graphs is geometric and yields a partial geometry with parameters pg(5,5,2) that is not isomorphic to the partial geometry discovered by J. H. van Lint and A. Schrijver in 1981.

math.CO↗
arXiv2019

On the PSU(4, 2)-invariant vertex-transitive strongly regular (216, 40, 4, 8) graph

In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $Γ$ with parameters (216, 40, 4, 8) and proved that it is the unique PSU(4,2)-invariant vertex-transitive graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of PG(3, 4), we provide a computer-free proof of the existence of the graph $Γ$. The maximal cliques of $Γ$ are also determined.

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arXiv2016

Strongly regular graphs from orthogonal groups $O^+(6,2)$ and $O^-(6,2)$

In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of strongly regular graphs with parameters (216,40,4,8) and (540,187,58,68). We also construct a strongly regular graph with parameters (540,224,88,96) that was to the best of our knowledge previously unknown. Further, we show that under certain conditions an orbit matrix $M$ of a strongly regular graph $Γ$ can be used to define a new strongly regular graph $\widetildeΓ$, where the vertices of the graph $\widetildeΓ$ correspond to the orbits of $Γ$ (the rows of $M$). We show that some of the obtained graphs are related to each other in a way that one can be constructed from an orbit matrix of the other.

math.CO↗
arXiv2016

The Cameron-Liebler problem for sets

Cameron-Liebler line classes and Cameron-Liebler k-classes in PG(2k+1,q) are currently receiving a lot of attention. Links with the Erdős-Ko-Rado results in finite projective spaces occurred. We introduce here in this article the similar problem on Cameron-Liebler classes of sets, and solve this problem completely, by making links to the classical Erdős-Ko-Rado result on sets. We also present a characterisation theorem for the Cameron-Liebler classes of sets.

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