On the intersection density of primitive groups of degree a product of two odd primes

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $\omega \in \Omega$ such that $\omega^g = \omega^h$. The \emph{intersection density} $\rho(G)$ of $G$ is the maximum of \left\{ \frac{|\mathcal{F}|}{|G_\omega|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}, where $G_\omega$ is the stabilizer of $\omega$ in $G$. In this paper, it is proved that if $G$ is an imprimitive group of degree $pq$, where $p$ and $q$ are distinct odd primes, with at least two systems of imprimitivity then $\rho(G) = 1$. Moreover, if $G$ is primitive of degree $pq$, where $p$ and $q$ are distinct odd primes, then it is proved that $\rho(G) = 1$, whenever the socle of $G$ admits an imprimitive subgroup.Comment: 22 pages, a new section was added. Accepted in Journal of Combinatorial Theory, Series

On cocliques in commutative Schurian association schemes of the symmetric group

Given the symmetric group $G = \operatorname{Sym}(n)$ and a multiplicity-free subgroup $H\leq G$, the orbitals of the action of $G$ on $G/H$ by left multiplication induce a commutative association scheme. The irreducible constituents of the permutation character of $G$ acting on $G/H$ are indexed by partitions of $n$ and if $\lambda \vdash n$ is the second largest partition in dominance ordering among these, then the Young subgroup $\operatorname{Sym}(\lambda)$ admits two orbits in its action on $G/H$, which are $\mathcal{S}_\lambda$ and its complement. In their monograph [Erd\H{o}s-Ko-Rado theorems: Algebraic Approaches. {\it Cambridge University Press}, 2016] (Problem~16.13.1), Godsil and Meagher asked whether $\mathcal{S}_\lambda$ is a coclique of a graph in the commutative association scheme arising from the action of $G$ on $G/H$. If such a graph exists, then they also asked whether its smallest eigenvalue is afforded by the $\lambda$-module. In this paper, we initiate the study of this question by taking $\lambda = [n-1,1]$. We show that the answer to this question is affirmative for the pair of groups $\left(G,H\right)$, where $G = \operatorname{Sym}(2k+1)$ and $H = \operatorname{Sym}(2) \wr \operatorname{Sym}(k)$, or $G = \operatorname{Sym}(n)$ and $H$ is one of $\operatorname{Alt}(k) \times \operatorname{Sym}(n-k),\ \operatorname{Alt}(k) \times \operatorname{Alt}(n-k)$, or $\left(\operatorname{Alt}(k)\times \operatorname{Alt}(n-k)\right) \cap \operatorname{Alt}(n)$. For the pair $(G,H) = \left(\operatorname{Sym}(2k),\operatorname{Sym}(k)\wr \operatorname{Sym}(2)\right)$, we also prove that the answer to this question of Godsil and Meagher is negative

Intersection density of imprimitive groups of degree pqpq

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on an element of $\Omega$. The \emph{intersection density} of $G$ is the number \rho(G) = \max\left\{ \frac{\mathcal{|F|}}{|G|/|\Omega|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}. Recently, Hujdurovi\'{c} et al. [Finite Fields Appl., 78 (2022), 101975] disproved a conjecture of Meagher et al. (Conjecture~6.6~(3) in [ J.~Combin. Theory, Ser. A 180 (2021), 105390]) by constructing equidistant cyclic codes which yield transitive groups of degree $pq$, where $p = \frac{q^k-1}{q-1}$ and $q$ are odd primes, and whose intersection density equal to $q$. In this paper, we use the cyclic codes given by Hujdurovi\'{c} et al. and their permutation automorphisms to construct a family of transitive groups $G$ of degree $pq$ with $\rho(G) = \frac{q}{k}$, whenever $k<q<p=\frac{q^k-1}{q-1}$ are odd primes. Moreover, we extend their construction using cyclic codes of higher dimension to obtain a new family of transitive groups of degree a product of two odd primes $q<p = \frac{q^k-1}{q-1}$, and whose intersection density are equal to $q$. Finally, we prove that if $G\leq \operatorname{Sym}(\Omega)$ of degree a product of two arbitrary odd primes $p>q$ and $\left\langle \bigcup_{\omega\in \Omega} G_\omega \right\rangle$ is a proper subgroup, then $\rho(G) \in \{1,q\}$

The qq-Analogue of Zero Forcing for Certain Families of Graphs

Zero forcing is a combinatorial game played on a graph with the ultimate goal of changing the colour of all the vertices at minimal cost. Originally this game was conceived as a one player game, but later a two-player version was devised in-conjunction with studies on the inertia of a graph, and has become known as the $q$-analogue of zero forcing. In this paper, we study and compute the $q$-analogue zero forcing number for various families of graphs. We begin with by considering a concept of contraction associated with trees. We then significantly generalize an equation between this $q$-analogue of zero forcing and a corresponding nullity parameter for all threshold graphs. We close by studying the $q$-analogue of zero forcing for certain Kneser graphs, and a variety of cartesian products of structured graphs.Comment: 29 page

On the intersection density of the Kneser Graph K(n,3)K(n,3)

A set $\mathcal{F} \subset \operatorname{Sym}(V)$ is \textsl{intersecting} if any two of its elements agree on some element of $V$. Given a finite transitive permutation group $G\leq \operatorname{Sym}(V)$, the \textsl{intersection density} $\rho(G)$ is the maximum ratio $\frac{|\mathcal{F}||V|}{|G|}$ where $\mathcal{F}$ runs through all intersecting sets of $G$. The \textsl{intersection density} $\rho(X)$ of a vertex-transitive graph $X = (V,E)$ is equal to \max \left\{ \rho(G) : G \leq \operatorname{Aut}(X) \mbox{ is transitive} \right\}. In this paper, we study the intersection density of the Kneser graph $K(n,3)$, for $n\geq 7$. The intersection density of $K(n,3)$ is determined whenever its automorphism group contains $\operatorname{PSL}_{2}(q)$ or $\operatorname{PGL}_{2}(q)$, with some exceptional cases depending on the congruence of $q$.Comment: 15 page

Infinite families of vertex-transitive graphs with prescribed Hamilton compression

Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $\kappa(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $\kappa(X)$ of $X$ } is the maximum of $\kappa(X,C)$ where $C$ runs over all Hamilton cycles in $X$. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor, Merino and M\"utze in [``The Hamilton compression of highly symmetric graphs'', {\em arXiv preprint} arXiv: 2205.08126v1 (2022)] whether for every positive integer $k$ there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to $k$. Since an infinite family of Cayley graphs with Hamilton compression equal to $1$ was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products $\mathbb{Z}_p\rtimes\mathbb{Z}_k$ where $p$ is a prime and $k \geq 2$ a divisor of $p-1$. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to $1$ are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given.Comment: 11 page