arXiv subjects

Roghayeh Maleki

Publications and source records attributed to Roghayeh Maleki.

27 records · Page 2Linked to original sources

arXiv2023

The $q$-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.

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arXiv2023

Distance-regular graphs with classical parameters that support a uniform structure: case $q \le 1$

Let $Γ=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid \partial(x,y) = \partial(x,z)\}$, where $\partial$ denotes the path-length distance in $Γ$. Observe that the graph $Γ_f=(X,\mathcal{R}_f)$ is bipartite. We say that $Γ$ supports a uniform structure with respect to $x$ whenever $Γ_f$ has a uniform structure with respect to $x$. Assume that $Γ$ is a distance-regular graph with classical parameters $(D,q,α,β)$ with $q \le 1$. Recall that $q$ is an integer, which is not equal to $0$ or $-1$. The purpose of this paper is to study when $Γ$ supports a uniform structure with respect to $x$. The main result of the paper is a complete classification of graphs with classical parameters with $q\leq 1$ and $D \ge 4$ that support a uniform structure with respect to $x$.

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arXiv2022

On closed distance magic circulants of valency up to $5$

Let $Γ=(V,E)$ be a graph of order $n$. A {\em closed distance magic labeling} of $Γ$ is a bijection $\ell : V \to \{1,2, \ldots, n\}$ for which there exists a positive integer $r$ such that $\sum_{x \in N[u]} \ell(x) = r$ for all vertices $u \in V$, where $N[u]$ is the closed neighborhood of $u$. A graph is said to be {\em closed distance magic} if it admits a closed distance magic labeling. In this paper, we classify all connected closed distance magic circulants with valency at most $5$, that is, Cayley graphs $\operatorname{Cay}(\mathbb{Z}_n;S)$ where $|S| \le 5$ and $S$ generates $\mathbb{Z}_n$.

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arXiv2022

The search for small association schemes with noncyclotomic eigenvalues

In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic variety for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. Our main results show the eigenvalues of commutative association schemes of rank 4 and nonsymmetric commutative association schemes of rank 5 will always be cyclotomic. In the rank 5 cases these our conclusions rely on calculations done by computer for Gröbner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.

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arXiv2022

On the intersection density of the symmetric group acting on uniform subsets of small size

Given a finite transitive group $G\leq \operatorname{Sym}(Ω)$, a subset $\mathcal{F}$ of $G$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on some element of $Ω$. The \emph{intersection density} of $G$, denoted by $ρ(G)$, is the maximum of the rational number $|\mathcal{F}|\left(\frac{|G|}{|Ω|}\right)^{-1}$ when $\mathcal{F}$ runs through all intersecting sets in $G$. In this paper, we prove that if $G$ is the group $\operatorname{Sym}(n)$ or $\operatorname{Alt}(n)$ acting on the $k$-subsets of $\{1,2,3\ldots,n\}$, for $k\in \{3,4,5\}$, then $ρ(G)=1$. Our proof relies on the representation theory of the symmetric group and the ratio bound.

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arXiv2021

No Hilton-Milner type results for linear groups of degree two

A set of permutations $\mathcal{F}$ of a finite transitive permutation group $G\leq \operatorname{Sym}(Ω)$ is \emph{intersecting} if any pair of elements of $\mathcal{F}$ agree on an element of $Ω$. We say that $G$ has the \emph{EKR property} if an intersecting set of $G$ has size at most the order of a point stabilizer. Moreover, $G$ has the \emph{strict-EKR} property whenever $G$ has the EKR property and any intersecting set of maximum size is a coset of a point stabilizer of $G$. It is known that the permutation group $\operatorname{GL}_2(\mathbb{F}_q)$ acting on $Ω_q := \mathbb{F}_q^2\setminus\{0\}$ has the EKR property, but does not have the strict-EKR property since the stabilizer of a hyperplane is a maximum intersecting set. In this paper, it is proved that the Hilton-Milner type result does not hold for $\operatorname{GL}_2(\mathbb{F}_q)$ acting on $Ω_q$. Precisely, it is shown that a maximal intersecting set of $\operatorname{GL}_2(\mathbb{F}_q)$ is of maximum size. As a result, we prove the Complete Erdős-Ko-Rado theorem for $\operatorname{GL}_2(\mathbb{F}_{q})$.

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arXiv2021

On non-normal subgroup perfect codes

Let $X = (V,E)$ be a graph. A subset $C \subseteq V(X)$ is a \emph{perfect code} of $X$ if $C$ is a coclique of $X$ with the property that any vertex in $V(X)\setminus C$ is adjacent to exactly one vertex in $C$. Given a finite group $G$ with identity element $e$ and $H\leq G$, $H$ is a \emph{subgroup perfect code} of $G$ if there exists an inverse-closed subset $S \subseteq G\setminus \{e\}$ such that $H$ is a perfect code of the Cayley graph $\operatorname{Cay}(G,S)$ of $G$ with connection set $S$. In this short note, we give an infinite family of finite groups $G$ admitting a non-normal subgroup perfect code $H$ such that there exists $ g\in G$ with $g^2\in H$ but $(gh)^2 \neq e$, for all $h \in H$; thus, answering a question raised by Wang, Xia, and Zhou in [Perfect sets in Cayley graphs. {\it arXiv preprint} arXiv:2006.05100, 2020].

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arXiv2021

$3$-setwise intersecting families of the symmetric group

Given two positive integers $n\geq 3$ and $t\leq n$, the permutations $σ,π\in \operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\ldots,n\}$. A family $\mathcal{F} \subset \operatorname{Sym}(n)$ is $t$-setwise intersecting if any two permutations of $\mathcal{F}$ are $t$-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if $t\leq n$ and $\mathcal{F} \subset \operatorname{Sym}(n)$ is a $t$-setwise intersecting family, then $|\mathcal{F}|\leq t!(n-t)!$ and equality holds only if $\mathcal{F}$ is a coset of a setwise stablizer of a $t$-subset of $\{1,2,\ldots,n\}$. In this paper, we prove that if $n\geq 11$ and $\mathcal{F}$ is $3$-setwise intersecting, then $|\mathcal{F}|\leq 6(n-3)!$. Moreover, we prove that the characteristic vector of a $3$-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of $\operatorname{Sym}(n)$ acting on the $3$-subsets of $\{1,2,\ldots,n\}$.

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arXiv2021

On the second eigenvalue of a Cayley graph of the symmetric group

In 2020, Siemons and Zalesski [On the second eigenvalue of some Cayley graphs of the symmetric group. {\it arXiv preprint arXiv:2012.12460}, 2020] determined the second eigenvalue of the Cayley graph $Γ_{n,k} = \operatorname{Cay}(\operatorname{Sym}(n), C(n,k))$ for $k = 0$ and $k=1$, where $C(n,k)$ is the conjugacy class of $(n-k)$-cycles. In this paper, it is proved that for any $n\geq 3$ and $k\in \mathbb{N}$ relatively small compared to $n$, the second eigenvalue of $Γ_{n,k}$ is the eigenvalue afforded by the irreducible character of $\operatorname{Sym}(n)$ that corresponds to the partition $[n-1,1]$. As a byproduct of our method, the result of Siemons and Zalesski when $k \in \{0,1\}$ is retrieved. Moreover, we prove that the second eigenvalue of $Γ_{n,n-5}$ is also equal to the eigenvalue afforded by the irreducible character of the partition $[n-1,1]$.

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