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Dean Crnković

Publications and source records attributed to Dean Crnković.

18 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.

math.CO↗
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.

math.CO↗
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.

math.CO↗
arXiv2022

Self-dual Hadamard bent sequences

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application ( Solé et al, 2021). We study the self dual class in length at most $196.$ We use three competing methods of generation: Exhaustion, Linear Algebra and Groebner bases. Regular Hadamard matrices and Bush-type Hadamard matrices provide many examples. We conjecture that if $v$ is an even perfect square, a self-dual bent sequence of length $v$ always exist. We introduce the strong automorphism group of Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group.

math.CO↗
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↗
arXiv2021

Neighbour-Transitive Codes and Partial Spreads in Generalised Quadrangles

A code $C$ in a generalised quadrangle ${\mathcal Q}$ is defined to be a subset of the vertex set of the point-line incidence graph $\varGamma$ of ${\mathcal Q}$. The minimum distance $δ$ of $C$ is the smallest distance between a pair of distinct elements of $C$. The graph metric gives rise to the distance partition $\{C,C_1,\ldots,C_ρ\}$, where $ρ$ is the maximum distance between any vertex of $\varGamma$ and its nearest element of $C$. Since the diameter of $\varGamma$ is $4$, both $ρ$ and $δ$ are at most $4$. If $δ=4$ then $C$ is a partial ovoid or partial spread of ${\mathcal Q}$, and if, additionally, $ρ=2$ then $C$ is an ovoid or a spread. A code $C$ in ${\mathcal Q}$ is neighbour-transitive if its automorphism group acts transitively on each of the sets $C$ and $C_1$. Our main results i) classify all neighbour-transitive codes admitting an insoluble group of automorphisms in thick classical generalised quadrangles that correspond to ovoids or spreads, and ii) give two infinite families and six sporadic examples of neighbour-transitive codes with minimum distance $δ=4$ in the classical generalised quadrangle ${\mathsf W}_3(q)$ that are not ovoids or spreads.

math.CO↗
arXiv2020

On automorphism groups of a biplane (121,16,2)

The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$.

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.

math.CO↗
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.

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