Eric Swartz's research while affiliated with College of William and Mary and other places

Publications (41)

Preprint
We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc...
Preprint
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A cover of an associative (not necessarily commutative nor unital) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $\sigma$-elementary if $\sigma(R) < \sigma(R/I)$ for every non...
Article
A cover of a unital, associative (not necessarily commutative) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ(R) of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if σ(R)<σ(R/I) for every nonzero two-sided ideal I of R. In this paper...
Article
A partial difference set S in a finite group G satisfying 1 ∉ S and S = S − 1 corresponds to an undirected strongly regular Cayley graph Cay ( G , S ) . While the case when G is abelian has been thoroughly studied, there are comparatively few results when G is nonabelian. In this paper, we provide restrictions on the parameters of a partial differe...
Preprint
A cover of a unital, associative (not necessarily commutative) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $\sigma$-elementary if $\sigma(R) < \sigma(R/I)$ for every nonzero...
Article
Ostrom and Wagner (1959) proved that if the automorphism group G G of a finite projective plane π \pi acts 2 2 -transitively on the points of π \pi , then π \pi is isomorphic to the Desarguesian projective plane and G G is isomorphic to P Γ L ( 3 , q ) \mathrm {P} \Gamma \mathrm {L}(3,q) (for some prime-power q q ). In the more general case of a fi...
Preprint
A partial difference set $S$ in a finite group $G$ satisfying $1 \notin S$ and $S = S^{-1}$ corresponds to an undirected Cayley graph ${\rm Cay}(G,S)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial diff...
Article
Nearly 60 years ago, László Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of...
Article
An n×n matrix H is Butson–Hadamard if its entries are kth roots of unity and it satisfies HH∗=nIn. Write BH(n,k) for the set of such matrices. Suppose that k=pαqβ where p and q are primes and α≥1. A recent result of Östergård and Paavola uses a matrix H∈BH(n,pk) to construct H′∈BH(pn,k). We simplify the proof of this result and remove the restricti...
Article
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A k-arc in the projective space \(\mathrm{PG}(n,q)\) is a set of k projective points such that no subcollection of \(n+1\) points is contained in a hyperplane. In this paper, we construct new 60-arcs and 110-arcs in \(\mathrm{PG}(4,q)\) that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set \(\...
Article
Full-text available
The covering number of a group G, denoted by σ(G), is the size of a minimal collection of proper subgroups of G whose union is G. We investigate which integers are covering numbers of groups. We determine which integers 129 or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group w...
Preprint
Full-text available
An n × n matrix H is Butson-Hadamard if its entries are k th roots of unity and it satisfies HH * = nI n. Write BH(n, k) for the set of such matrices. Suppose that k = p α q β where p and q are primes and α ≥ 1. A recent result of¨Ostergårdof¨ of¨Ostergård and Paavola uses a matrix H ∈ BH(n, pk) to construct H ∈ BH(pn, k). We simplify the proof of...
Preprint
Full-text available
An $n \times n$ matrix $H$ is Butson-Hadamard if its entries are $k^{\text{th}}$ roots of unity and it satisfies $HH^* = nI_n$. Write $\textrm{BH}(n, k)$ for the set of such matrices. Suppose that $k = p^{\alpha}q^{\beta}$ where $p$ and $q$ are primes and $\alpha \geq 1$. A recent result of {\"O}sterg{\aa}rd and Paavola uses a matrix $H \in \textrm...
Preprint
Nearly $60$ years ago, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group o...
Preprint
Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $\pi$ acts $2$-transitively on the points of $\pi$, then $\pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{P\Gamma L}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible sph...
Article
Full-text available
In a User-Private Information Retrieval (UPIR) scheme, a set of users collaborate to retrieve files from a database without revealing to observers which participant in the scheme requested the file. Protocols have been proposed based on pairwise balanced designs and symmetric designs. Wepropose a new class of UPIR schemes based on generalised quadr...
Preprint
In this paper, we study prime order automorphisms of generalized quadrangles. We show that, if $\mathcal{Q}$ is a thick generalized quadrangle of order $(s,t)$, where $s > t$ and $s+1$ is prime, and $\mathcal{Q}$ has an automorphism of order $s+1$, then \[ s \left\lceil \left\lceil \frac{t^2}{s+1}\right\rceil\left(\frac{s+1}{t} \right) \right\rceil...
Preprint
A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{\ast} = nI_{n}$. We write $\mathrm{BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\mathrm{BH}(n, k) \rightarrow \mathrm{BH}(m, \ell)$. In this paper, we develop a technique for controlling the spectra...
Preprint
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If a group $G$ is the union of proper subgroups $H_1, \dots, H_k$, we say that the collection $\{H_1, \dots, H_k \}$ is a cover of $G$, and the size of a minimal cover (supposing one exists) is the covering number of $G$, denoted by $\sigma(G)$. The aim of this paper is to investigate which integers are covering numbers of groups. Until now, it has...
Preprint
A \textit{k}-arc in the projective space ${\rm PG}(n,q)$ is a set of $k$ projective points such that no subcollection of $n+1$ points is contained in a hyperplane. Given a set $\mathcal{P}$ of $k$ projective points of ${\rm PG}(n,\mathbb{C})$, we introduce computational methods in ${\rm GAP}$ that are effective for verifying both the existence of $...
Article
A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we show t...
Article
Let $R$ be an associative commutative ring with unity and $M_n(R)$ the ring of $n \times n$ matrices over $R$. A zero pattern matrix ring is a subring of $M_n(R)$ defined by the location of zero and nonzero entries of matrices in the subring. It is known that the set of zero pattern matrix rings of $M_n(R)$ is in bijective correspondence with the s...
Article
Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching $\mathcal{M}$ setwise allows the edges of $\mathcal{M}$ to be permuted in any fashion. A matching $\mathcal{M}$...
Article
If a group is the union of proper subgroups , we say that the collection is a cover of , and the size of a minimal cover (supposing one exists) is the covering number of , denoted by . Maróti showed that for odd and sufficiently large, and he also gave asymptotic bounds for even. In this paper, we determine the exact value of when is divisible by s...
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Full-text available
In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space (Formula presented.), unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems. Reproving these results has allowed us to find...
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Full-text available
In this paper we investigate a parameter of graphs, called the circular altitude, introduced by Peter Cameron. We show that the circular altitude provides a lower bound on the circular chromatic number, and hence on the chromatic number, of a graph and investigate this parameter for the iterated Mycielskian of certain graphs.
Article
We show that there exist functions c and g such that, if k, n and d are positive integers with d>g(n) and Γ is a d-valent 2-arc-transitive graph of order kpn with p a prime, then p≤kc(d). In other words, there are only finitely many d-valent 2-arc-transitive graphs of order kpn with d>g(n) and p prime. This generalises a recent result of Conder, Li...
Article
A generalized quadrangle is a point-line incidence geometry $\mathcal{Q}$ such that: (i) any two points lie on at most one line, and (ii) given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. The finite Moufang generalized quadrangles were classified by Fong and Seitz (1973), and we stud...
Article
In this paper, we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space $\mathrm{H}(3, q^2 )$, unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems.
Article
We show that there exist functions $c$ and $g$ such that, if $k$, $n$ and $d$ are positive integers with $d> g(n)$ and $\Gamma$ is a $d$-valent $2$-arc-transitive graph of order $kp^n$ with $p$ a prime, then $p\leqslant kc(d)$. In other words, there are only finitely many $d$-valent 2-arc-transitive graphs of order $kp^n$ with $d>g(n)$ and $p$ prim...
Article
If a group $G$ is the union of proper subgroups $H_1, ..., H_k$, we say that the collection $\{H_1, ...H_k \}$ is a cover of $G$, and the size of a minimal cover (supposing one exists) is the covering number of $G$, denoted $\sigma(G)$. Mar\'{o}ti showed that $\sigma(S_n) = 2^{n-1}$ for $n$ odd and sufficiently large, and he also gave asymptotic bo...
Article
A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover $G$ is called the covering number of $G$, denoted by $\sigma(G)$. Determining $\sigma(G)$ is an open problem for many non-solvable groups. For symmetric groups $S_n$, Mar\'oti determined $\sigma(S_n)$ for odd $n$ with...
Article
The only known skew-translation generalised quadrangles (STGQ) having order $(q,q)$, with $q$ even, are translation generalised quadrangles. Equivalently, the only known groups $G$ of order $q^3$, $q$ even, admitting an Ahrens-Szekeres (AS-)configuration are elementary abelian. In this paper we prove results in the theory of STGQ giving (i) new str...
Article
A partial difference set (Formula presented.) in a finite group (Formula presented.) such that (Formula presented.) and (Formula presented.) corresponds to an undirected strongly regular Cayley graph (Formula presented.) Very few examples of PDSs are known, and there are especially few known in nonabelian groups. In this paper, a partial difference...
Article
In this paper, seven families of vertex-intransitive locally (G,2)(G,2)-arc transitive graphs are constructed, where Sz(q)⩽G⩽Aut(Sz(q))Sz(q)⩽G⩽Aut(Sz(q)), q=22k+1q=22k+1 for some k∈Nk∈N. It is then shown that for any graph Γ in one of these families, Sz(q)⩽Aut(Γ)⩽Aut(Sz(q))Sz(q)⩽Aut(Γ)⩽Aut(Sz(q)) and that the only locally 2-arc transitive graphs ad...
Article
A near-polygonal graph is a graph Γ which has a set of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in . If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of valency r and girth m, Archdeacon and Perkel proved the existence of a polygonal graph Γ2 of valency r a...
Article
A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal. We provide a construction of an infinite family of polygonal graphs of arbitrary odd girth with 2-arc transitive automorphism grou...
Article
A near-polygonal graph is a graph Γ which has a set \({\mathcal{C}}\) of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in \({\mathcal{C}}\) . If m is the girth of Γ then the graph is called polygonal. We provide a construction for an infinite family of 2-arc transitive near-polygonal graphs of val...

Citations

... Indeed, Kappe [18, p. 87] further writes, "An interesting question would be if there are integers n > 2 that are not the covering number of a ring." Other recent works on this problem include [5,9,27,32,34]. ...
... In recent years, there has been significant progress towards Kantor's 1991 conjecture [18] that a finite flag-transitive generalized quadrangle is either classical or has order (3,5) or (15,17) up to duality, cf. [3,4]. The generalized quadrangles with an automorphism group acting primitively on points have also attracted much attention, cf. ...
... The question of which groups come up as unit groups of k-algebras can also be approached using results on which groups come up as unit groups of rings, also known as Fuchs' Problem. Though to combine results with Sakurai's criterion one would rather need to study normalized unit groups, which need to be properly defined for general rings, but at least in characteristic 2 results such as [SW20] 4 could be directly applied. ...
... [26, 2.4.1], [1,Lemma 2.5]) Let g be an automorphism of a generalized quadrangle S = (P, B, I) of order (s, t). Let P g and B g be the set of fixed points and fixed lines of g respectively, and set I g := I ∩ P g × B g , S g := (P g , B g , I g ). ...
... This motives the study of Butson matrices even if real Hadamard matrices are the primary interest. In Section 2.2 we construct a morphism BH(n, k) → BH(nm, k/m) where k = p e 1 1 · · · p et t and m = p e 1 −1 1 · · · p et−1 t , matching the parameters of the morphism discovered by Ó Catháin and Swartz in [18]. But their applications in applied sciences most strongly motivate their study. ...
... As noted above, no group has covering number 7, and other natural numbers exist that are not the covering number of a group (the next smallest examples being 11 [10] and 19 [14]). All the integers N 129 that are not the covering number of a group were determined in [15], which also contains a good summary of the history and recent development of the related work for groups. It is not known whether there are infinitely many positive integers that are not the covering number of a group. ...
... In case n is divisible by 6, E. Swartz [20] managed to give a formula for σ(S n ). Apart from this case, the value of σ(S n ) for n even is only known for n ≤ 14: see [2,14,18]. One objective of the present paper is to determine σ(S n ) for every even integer n ≥ 26. ...
... The study of finite generalized quadrangles with a regular automorphism group was initiated by Ghinelli [12] in 1992, and we refer to [2,10,11,26,31] for research on this topic. By determining all the point regular automorphism groups of the Payne derived quadrangle of the symplectic quadrangle W (q), q odd, we [11] showed that the finite groups that act regularly on the points of a finite generalized quadrangle can have unbounded nilpotency class. ...
... Our introduction of permutable matchings led to the question: Which graphs have permutable matchings? That has been investigated by Schaefer and Swartz in [4]; they find large families of examples. On the other hand, there are only a few kinds of graph with permutable perfect matchings; Schaefer and Swartz determine them all. ...
... As a consequence, the clique number is a lower bound for the circular altitude. So, the inequality ω(G) ≤ α o (G) holds for every graph G. Also, since the circular altitude is a parameter related to monotonic cycles, it is a natural proposition that if α o (G) ≥ 3, then α o (G) is greater than or equal to the girth of G [1]. ...