arXiv subjects

Andriaherimanana Sarobidy Razafimahatratra

Publications and source records attributed to Andriaherimanana Sarobidy Razafimahatratra.

At least 19 recordsLinked to original sources

arXiv2026

The automorphism group of the derangement graph of $\operatorname{PGL}_{2}(q)$ acting on the projective line

Given a finite transitive group $G\leq \operatorname{Sym}(Ω)$, the derangement graph $Γ_G$ is the graph whose vertex set is $G$, and two vertices $g$ and $h$ are adjacent if the ratio $h^{-1}g$ is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of the transitive permutation group corresponding to the natural action of $\operatorname{PGL}_{2}(q)$ on the projective line $\operatorname{PG}_{1}(q)$ is \begin{align*} \operatorname{Aut}(Γ_{\operatorname{PGL}_{2}({q})}) = \left(L_{\operatorname{PGL}_2(q)}\times R_{\operatorname{PGL}_2(q)}\right) \rtimes \left(\langle ψ\rangle \times γ_{\operatorname{Aut}(\mathbb{F}_q)}\right), \end{align*} where $L_{\operatorname{PGL}_2(q)}$ is the left-regular representation of $\operatorname{PGL}_2(q)$, $R_{\operatorname{PGL}_2(q)}$ is the right-regular representation of $\operatorname{PGL}_2(q)$, $γ_{\operatorname{Aut}(\mathbb{F}_q)}$ is the group of conjugation by elements of $\operatorname{Aut}(\mathbb{F}_q)$, and $ψ: \operatorname{PGL}_2(q) \to \operatorname{PGL}_2(q)$ such that $ψ(x) = x^{-1}$.

math.CO↗

arXiv2026

On the Terwilliger algebras of quasi-thin Schurian association schemes

An association scheme is triply transitive if its automorphism group is transitive, and the centralizer algebra of a point stabilizer of its automorphism group coincides with the Terwilliger algebra and its subspace $T_0$. In this paper, we give necessary and sufficient conditions for a quasi-thin association scheme to be triply transitive. As a by-product, we give new infinite families of triply-transitive association schemes.

math.CO↗

arXiv2026

Perfect and multiple state transfer in oriented Cayley graphs

We study perfect state transfer and multiple state transfer in oriented normal Cayley graphs. We construct examples in a variety of groups, ranging from abelian to nonsolvable, and establish some general restrictions and nonexistence results.

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arXiv2025

The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers

Given a finite transitive group $G\leq \operatorname{Sym}Ω$, the {intersection density} of $G$ is defined as the ratio between the size of the largest subsets of $G$ in which any two permutations agree on at least one element of $Ω$, and the order of a point stabilizer of $G$. In this paper, we completely determine the intersection densities of the permutation groups $\operatorname{PSL}_{2}(q)$, where $q$ is a power of an odd prime $p$, acting transitively with point stabilizers conjugate to $\mathbb{Z}_p$. Our proof uses an auxiliary graph, which is a $\operatorname{PGL}_{2}{q}$-vertex-transitive graph, in which a clique corresponds to an intersecting set of $\operaotnrame{PSL}_{2}(q)$. For the transitive action of $\psl{2}{q}$ with point stabilizers conjugate to $\mathbb{Z}_r$, where $r\mid \frac{q-1}{2}$ is an odd prime, we show that the auxiliary graph is not regular, and we construct an intersecting set which is sometimes of maximum size.

math.CO↗

arXiv2025

On the classification of triply-transitive strongly-regular graphs

Let $Γ= (Ω,E)$ be a strongly-regular graph with adjacency matrix $A_1$, and let $A_2$ be the adjacency matrix of its complement. For any vertex $ω\in Ω$, we define $E_{0,ω}^*$ $E_{1,ω}^*$ and $E_{2,ω}^*$ to be respectively the diagonal matrices whose main diagonal is the row corresponding to $ω$ in the matrices $I, A_1$, and $A_2$. The Terwilliger algebra of $Γ$ with respect to the vertex $ω\in Ω$ is the subalgebra $T_ω= \left\langle I,A_1,A_2,E_{0,ω}^*,E_{1,ω}^*,E_{2,ω}^* \right\rangle$ of the complex matrix algebra $\operatorname{M_{|Ω|}}(\mathbb{C})$. The algebra $T_ω$ contains the subspace $T_{0,ω} = \operatorname{Span}\left\{ E_{i,ω}^*A_jE_{k,ω}^*: 0\leq i,j,k\leq 2 \right\}$. In addition, if $G = \AutΓ$, then $T_ω$ is a subalgebra of the centralizer algebra $\tilde{T}_ω= \End{G_ω}{\mathbb{C}^Ω}$. The strongly-regular graph $Γ=(Ω,E)$ is triply transitive if $Γ$ is vertex transitive and $T_{0,ω} = T_ω= \tilde{T}_ω$, for any $ω\in Ω$. In this paper, we classify all triply transitive strongly-regular graphs that are not isomorphic to the collinearity graph of the polar space $O_{6}^-(q)$, where $q$ is a prime power, or the affine polar graph $\vo_{2m}^\varepsilon(2)$, where $m\geq 1$ and $\varepsilon = \pm 1$.

math.CO↗

arXiv2025

The intersection density of cubic arc-transitive graphs with $2$-arc-regular full automorphism group equal to $\operatorname{PGL}_2(q)$

The \emph{intersection density} of a transitive permutation group $G\leq \operatorname{Sym}(Ω)$ is the ratio between the largest size of a subset of $G$ in which any two agree on at least one element of $Ω$, and the order of a point-stabilizer of $G$. In this paper, we determine the intersection densities of the automorphism group of the arc-transitive graphs admitting a $2$-arc-regular full automorphism group $G^* = \operatorname{PGL}_2(q)$ and an arc-regular subgroup of automorphism $G = \operatorname{PSL}_2(q)$.

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arXiv2024

On the Terwilliger algebra of the group association scheme of the symmetric group $\operatorname {sym}(7)$

Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $\operatorname {sym}(n)$, for $3\leq n \leq 6$, have been studied and completely determined. The case for $\operatorname {sym}(7)$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $\operatorname {sym}(7)$ with a minimal distance of at least $4$. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $\operatorname {sym}(7)$ are determined.

math.CO↗

arXiv2024

On the maximum intersecting sets of the general semilinear group of degree $2$

Let $p$ be a prime and $q = p^k$. A subset $\mathcal{F} \subset \operatorname{ΓL}_{2}(q)$ is intersecting if any two semilinear transformations in $\mathcal{F}$ agree on some non-zero vector in $\mathbb{F}_q^2$. We show that any intersecting set of $\operatorname{ΓL}_{2}(q)$ is of size at most that of a stabilizer of a non-zero vector, and we characterize the intersecting sets of this size. Our proof relies on finding a subgraph which is a lexicographic product in the derangement graph of $\operatorname{ΓL}_{2}(q)$ in its action on non-zero vectors of $\mathbb{F}_q^2$. This method is also applied to give a new proof that the only maximal intersecting sets of $\operatorname{GL}_{2}(q)$ are the maximum intersecting sets.

math.CO↗

arXiv2023

Intersection density of imprimitive groups of degree $pq$

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(Ω)$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on an element of $Ω$. The \emph{intersection density} of $G$ is the number $$ρ(G) = \max\left\{ \mathcal{|F|}/|G_ω| \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\},$$ where $ω\inΩ$ and $G_ω$ is the stabilizer of $ω$ in $G$. It is known that if $G\leq \operatorname{Sym}(Ω)$ is an imprimitive group of degree a product of two odd primes $p>q$ admitting a block of size $p$ or two complete block systems, whose blocks are of size $q$, then $ρ(G) = 1$. In this paper, we analyse the intersection density of imprimitive groups of degree $pq$ with a unique block system with blocks of size $q$ based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than $1$ whenever there exists a cyclic code $C$ with parameters $[p,k]_q$ such that any codeword of $C$ has weight at most $p-1$, and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing $\operatorname{Alt}(5)$ or a projective special linear group. We give some examples where the latter has intersection density equal to $1$, under some restrictions on $p$ and $q$.

math.CO↗

arXiv2023

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

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} $ρ(G)$ is the maximum ratio $\frac{|\mathcal{F}||V|}{|G|}$ where $\mathcal{F}$ runs through all intersecting sets of $G$. The \textsl{intersection density} $ρ(X)$ of a vertex-transitive graph $X = (V,E)$ is equal to $\max \left\{ ρ(G) : G \leq \operatorname{Aut}(X), \mbox{ $G$ 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 $\operatorname{PSL}_{2}(q)$, with some exceptional cases depending on the congruence of $q$. We also briefly consider the intersection density of $K(n,2)$ for values of $n$ where $\operatorname{PSL}_{2}(q)$ is a subgroup of its automorphism group.

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arXiv2023

The Erdős-Ko-Rado Theorem for non-quasiprimitive groups of degree $3p$

The \emph{intersection density} of a finite transitive group $G\leq \operatorname{Sym}(Ω)$ is the rational number $ρ(G)$ given by the ratio between the maximum size of a subset of $G$ in which any two permutations agree on some elements of $Ω$ and the order of a point stabilizer of $G$. In 2022, Meagher asked whether $ρ(G)\in \{1,\frac{3}{2},3\}$ for any transitive group $G$ of degree $3p$, where $p\geq 5$ is an odd prime. For the primitive case, it was proved in [\emph{J. Combin. Ser. A}, 194:105707, 2023] that the intersection density is $1$. It is shown in this paper that the answer to this question is affirmative for non-quasiprimitive groups, unless possibly when $p = q+1$ is a Fermat prime and $Ω$ admits a unique $G$-invariant partition $\mathcal{B}$ such that the induced action $\overline{G}_\mathcal{B}$ of $G$ on $\mathcal{B}$ is an almost simple group containing $\operatorname{PSL}_{2}(q)$.

math.CO↗

arXiv2023

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 $λ\vdash n$ is the second largest partition in dominance ordering among these, then the Young subgroup $\operatorname{Sym}(λ)$ admits two orbits in its action on $G/H$, which are $\mathcal{S}_λ$ and its complement. In their monograph [Erdős-Ko-Rado theorems: Algebraic Approaches. {\it Cambridge University Press}, 2016] (Problem~16.13.1), Godsil and Meagher asked whether $\mathcal{S}_λ$ 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 $λ$-module. In this paper, we initiate the study of this question by taking $λ= [n-1,1]$. 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.

math.CO↗

arXiv2023

On the intersection spectrum of $\operatorname{PSL}_2(q)$

Given a group $G$ and a subgroup $H \leq G$, a set $\mathcal{F}\subset G$ is called $H$\emph{-intersecting} if for any $g,g' \in \mathcal{F}$, there exists $xH \in G/H$ such that $gxH=g'xH$. The \emph{intersection density} of the action of $G$ on $G/H$ by (left) multiplication is the rational number $ρ(G,H)$, equal to the maximum ratio $\frac{|\mathcal{F}|}{|H|}$, where $\mathcal{F} \subset G$ runs through all $H$-intersecting sets of $G$. The \emph{intersection spectrum} of the group $G$ is then defined to be the set $$ σ(G) := \left\{ ρ(G,H) : H\leq G \right\}. $$ It was shown by Bardestani and Mallahi-Karai [{\it J. Algebraic Combin.}, 42(1):111-128, 2015] that if $σ(G) = \{1\}$, then $G$ is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than $1$ belong to $σ(G)$, whenever $G$ is non-solvable. In this paper, we study the intersection spectrum of the linear group $\operatorname{PSL}_2(q)$. It is shown that $2 \in σ\left(\operatorname{PSL}_2(q)\right)$, for any prime power $q\equiv 3 \pmod 4$. Moreover, when $q\equiv 1 \pmod 4$, it is proved that $ρ(\operatorname{PSL}_2(q),H)=1$, for any odd index subgroup $H$ (containing $\mathbb{F}_q$) of the Borel subgroup (isomorphic to $\mathbb{F}_q\rtimes \mathbb{Z}_{\frac{q-1}{2}}$) consisting of all upper triangular matrices.

math.CO↗

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.

math.CO↗

arXiv2023

Infinite families of vertex-transitive graphs with prescribed Hamilton compression

Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $κ(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $κ(X)$ of $X$ } is the maximum of $κ(X,C)$ 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 Mütze 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.

math.CO↗

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 $\mathbb{Z}_n$.

math.CO↗

arXiv2022

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

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(Ω)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $ω\in Ω$ such that $ω^g = ω^h$. The \emph{intersection density} $ρ(G)$ of $G$ is the maximum of $\left\{ \frac{|\mathcal{F}|}{|G_ω|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}$, where $G_ω$ is the stabilizer of $ω$ 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 $ρ(G) = 1$. Moreover, if $G$ is primitive of degree $pq$, where $p$ and $q$ are distinct odd primes, then it is proved that $ρ(G) = 1$, whenever the socle of $G$ admits an imprimitive subgroup.

math.CO↗

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.

math.CO↗