arXiv subjects

Karen Meagher

Publications and source records attributed to Karen Meagher.

At least 19 recordsLinked to original sources

arXiv2026

Sedentary quantum walks on bipartite graphs

If a quantum walk starting on a vertex tends to stay at home, then that vertex is said to be sedentary. We prove that almost all planar graphs and almost all trees contain at least two sedentary vertices for any assignment of edge weights -- a result that suggests vertex sedentariness is a common phenomenon in trees and planar graphs. For weighted bipartite graphs, we show that a vertex is not sedentary whenever 0 does not belong to its eigenvalue support. Consequently, each vertex in a nonsingular weighted bipartite graph is not sedentary, a stark contrast to weighted trees and weighted planar graphs. A corollary of this result is that every vertex in a bipartite graph with a unique perfect matching is not sedentary for any assignment of edge weights. We also construct new families of weighted bipartite graphs with sedentary vertices using the bipartite double and subdivision operations. Finally, we show that unweighted paths and unweighted even cycles contain no sedentary vertices.

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arXiv2025

Hilton-Milner Theorem for the $r$-independent sets in a union of cliques

We give a Hilton-Milner Theorem for the $r$-independent sets in the graph that is the union of copies of $K_k$. That is, we determine the maximum intersecting families of $r$-independent sets in this graph, subject to the condition that the sets in a family do not all share a common element. As a by-product, we also find a tight upper bound for the sum of sizes of a pair of cross intersecting families made up of the same objects. We apply our theorem to find the largest intersecting family of $r$-independent sets in a family of graphs called ``depth-two claws". This confirms the Holroyd--Talbot conjecture for depth-two claws, extending previous results on these graphs (which covered cases where $r$ was relatively small compared to the number of vertices) to all possible values of $r$.

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arXiv2025

On the second largest eigenvalue of certain graphs in the perfect matching association scheme

The perfect matching association scheme is a set of relations on the perfect matchings of the complete graph on $2n$ vertices. The relations between perfect matchings are defined by the cycle structure of the union of any two perfect matchings, and each relation can be represented as a matrix. Each matrix is labeled by an integer partition whose parts correspond to the size do the cycles in the union. Since these matrices form an association scheme, they are simultaneously diagonalizable. Further, it is well-known that the common eigenspaces correspond to the irreducible representations of $S_{2n}$ indexed by the even partitions of $2n$. In this paper, we conjecture that the second largest eigenvalue of the matrices in the perfect matching association scheme labeled by a partition containing at least two parts of size 1 always occurs on the eigenspace corresponding to the representation indexed by $[2n-2, 2]$. We confirm this conjecture for matrices labeled by the partitions $[2, 1^{n-2}], [3, 1^{n-3}], [2, 2, 1^{n-4}], [4, 1^{n-4}], [3, 2, 1^{n-5}]$, and $[5, 1^{n-5}]$, as well as any partition in which the first part is sufficiently large.

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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 $G\leq \operatorname{Sym}(\Omega)$ is the ratio between the largest size of a subset of $G$ in which any two agree on at least one element of $\Omega$, 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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arXiv2025

A new measure of robustness of Erd\H{o}s--Ko--Rado Theorems on permutation groups

In this paper we introduce a new way of measuring the robustness of Erd\H{o}s--Ko--Rado (EKR) Theorems on permutation groups. EKR-type results can be viewed as results about the independence numbers of certain corresponding graphs, namely the derangement graphs, and random subgraphs of these graphs have been used to measure the robustness of these extremal results. In the context of permutation groups, the derangement graphs are Cayley graphs on the permutation group in question. We propose studying extremal properties of subgraphs of derangement graphs, that are themselves Cayley graphs of the group, to measure robustness. We present a variety of results about the robustness of the EKR property of various permutation groups using this new measure.

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arXiv2025

Intersecting Families of Spanning Trees

A family $\mathcal{F}$ of spanning trees of the complete graph on $n$ vertices $K_n$ is \emph{$t$-intersecting} if any two members have a forest on $t$ edges in common. We prove an Erd\H{o}s--Ko--Rado result for $t$-intersecting families of spanning trees of $K_n$. In particular, we show there exists a constant $C > 0$ such that for all $n \geq C (\log n) t$ the largest $t$-intersecting families are the families consisting of all trees that contain a fixed set of $t$ disjoint edges (as well as the stars on $n$ vertices for $t = 1$). The proof uses the spread approximation technique in conjunction with the Lopsided Lov\'asz Local Lemma.

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arXiv2024

Robustness of Erdős--Ko--Rado theorems on permutations and perfect matchings

The Erdős--Ko--Rado (EKR) theorem and its generalizations can be viewed as classifications of maximum independent sets in appropriately defined families of graphs, such as the Kneser graph $K(n,k)$. In this paper, we investigate the independence number of random spanning subraphs of two other families of graphs whose maximum independent sets satisfy an EKR-type characterization: the derangement graph on the set of permutations in $\mathrm{Sym}(n)$ and the derangement graph on the set $\mathcal{M}_{n}$ of perfect matchings in the complete graph $\mathcal{K}_{2n}$. In both cases, we show there is a sharp threshold probability for the event that the independence number of a random spanning subgraph is equal to that of the original graph. As a useful tool to aid our computations, we obtain a Friedgut--Kalai--Naor (FKN) type theorem on sparse boolean functions whose domain is the vertex set of $\mathcal{M}_{n}$. In particular, we show that boolean functions whose Fourier transforms are highly concentrated on the first two irreducible modules in the $\mathrm{Sym}(2n)$ module $\mathbb{C}[\mathcal{M}_{n}]$, is close to being the characteristic function of a union of maximum independent sets in the derangement graph on perfect matchings.

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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

Cameron-Liebler sets in permutation groups

Consider a group $G$ acting on a set $Ω$, the vector $v_{a,b}$ is a vector with the entries indexed by the elements of $G$, and the $g$-entry is 1 if $g$ maps $a$ to $b$, and zero otherwise. A $(G,Ω)$-Cameron-Liebler set is a subset of $G$, whose indicator function is a linear combination of elements in $\{v_{a, b}\ :\ a, b \in Ω\}$. We investigate Cameron-Liebler sets in permutation groups, with a focus on constructions of Cameron-Liebler sets for 2-transitive groups.

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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

Induced forests in some distance-regular graphs

In this article, we study the order and structure of the largest induced forests in some families of graphs. First we prove a variation of the ratio bound that gives an upper bound on the order of the largest induced forest in a graph. Next we define a \textsl{canonical induced forest} to be a forest that is formed by adding a vertex to a coclique and give several examples of graphs where the maximal forest is a canonical induced forest. These examples are all distance-regular graphs with the property that the Delsarte-Hoffman ratio bound for cocliques holds with equality. We conclude with some examples of related graphs where there are induced forests that are larger than a canonical forest.

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arXiv2022

Fusions of the generalized Hamming scheme on a strongly-regular graph

In this paper we show that for any fusion $\mathcal{B}$ of an association scheme $\mathcal{A}$, the generalized Hamming scheme $H(n,\mathcal{B})$ is a nontrivial fusion of $H(n,\mathcal{A})$. We analyze the case where $\mathcal{A}$ is the association scheme on a strongly-regular graph, and determine the parameters of all strongly-regular graphs for which the generalized Hamming scheme, $H(2,\mathcal{A})$, has extra fusions, in addition to the one arising from the trivial fusion of $\mathcal{A}$.

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arXiv2022

All $2$-transitive groups have the EKR-module property

We prove that every 2-transitive group has a property called the EKR-module property. This property gives a characterization of the maximum intersecting sets of permutations in the group. Specifically, the characteristic vector of any maximum intersecting set in a 2-transitive group is the linear combination of the characteristic vectors of the stabilizers of a points and their cosets. We also consider when the derangement graph of a 2-transitive group is connected and when a maximum intersecting set is a subgroup or a coset of a subgroup.

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arXiv2022

An upper bound for the $k$-power domination number in $r$-uniform hypergraphs

Generalizing work on graphs, Chang and Roussel introduced $k$-power domination in hypergraphs and conjectured the upper bound for the $k$-power domination number for $r$-uniform hypergraphs on $n$ vertices was $\frac{n}{r+k}$. This upper bound was shown to be true for simple graphs ($r=2$) and it was further conjectured that only a family of hypergraphs, known as the squid hypergraphs, attained this upper bound. In this paper, the conjecture is proven to hold for hypergraphs with $r=3$ or $4$; but is shown to be false, by a counterexample, for $r\geq 7$. Furthermore, we show that the squid hypergraphs are not the only hypergraphs that attain the original upper bound. Finally, a new upper bound is proven for $r\geq 3$.

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arXiv2021

Erdős-Ko-Rado results for the general linear group, the special linear group and the affine general linear group

In this paper, we show that both the general linear group $\gl{q}$ and the special linear group $\slg{q}$ have both the EKR property and the EKR-module property. This is done using an algebraic method; a weighted adjacency matrix for the derangement graph for the group is found and Hoffman's ratio bound is applied to this matrix. We also consider the group $\agl{q}$ and the 2-intersecting sets in $\PGL(2,q)$.

math.CO↗

arXiv2021

An Extension of the Erdős-Ko-Rado Theorem to uniform set partitions

A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left| P_{i} \cap Q_{j} \right|\geq t$. In this paper we prove a version of the Erdős-Ko-Rado theorem for partially $2$-intersecting $(k,\ell)$-partitions. In particular, we show for $\ell$ sufficiently large, the set of all $(k,\ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,\ell)$-partitions. For for $k=3$, we show this result holds for all $\ell$.

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arXiv2021

Ovoids of Generalized Quadrangles of Order $(q, q^2-q)$ and Delsarte Cocliques in Related Strongly Regular Graphs

We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcović's inertia bound are equal. This means that $ve^- = m^-(e^- - k)$, where $v$ is the number of vertices, $k$ is the regularity, $e^-$ is the smallest eigenvalue, and $m^-$ is the multiplicity of $e^-$. We show that Delsarte cocliques do not exist for all Taylor's $2$-graphs and for point graphs of generalized quadrangles of order $(q,q^2-q)$ for infinitely many $q$. For cases where equality may hold, we show that for nearly all parameter sets, there are at most two Delsarte cocliques.

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arXiv2020

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with partitions $[n-2,2]$ have been characterized. We find families of graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with multiplicity partition $[n-k,k]$ for $k\geq 2$. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in $\mathcal{S}(G)$ with multiplicity partition $[n-k,k]$ to show the complexities of characterizing these graphs.

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