arXiv subjects

Peter J. Cameron

Publications and source records attributed to Peter J. Cameron.

At least 91 records · Page 5Linked to original sources

arXiv2017

The cycle polynomial of a permutation group

The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $Ω$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples. In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is the orbital chromatic polynomial of $Γ$ and $G$, where $Γ$ is a $G$-invariant graph, introduced by the first author, Jackson and Rudd. We pose the general problem of finding all such reciprocal pairs, and give a number of examples and characterisations: the latter include the cases where $Γ$ is a complete or null graph or a tree. The paper concludes with some comments on other polynomials associated with a permutation group.

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arXiv2016

Primitive groups, road closures, and idempotent generation

We are interested in semigroups of the form $\langle G,a\rangle\setminus G$, where $G$ is a permutation group of degree $n$ and $a$ a non-permutation on the domain of $G$. A theorem of the first author, Mitchell and Schneider shows that, if this semigroup is idempotent-generated for all possible choices of $a$, then $G$ is the symmetric or alternating group of degree $n$, with three exceptions (having $n=5$ or $n=6$). Our purpose here is to prove stronger results where we assume that $\langle G,a\rangle\setminus G$ is idempotent-generated for all maps of fixed rank $k$. For $k\ge6$ and $n\ge2k+1$, we reach the same conclusion, that $G$ is symmetric or alternating. These results are proved using a stronger version of the \emph{$k$-universal transversal property} previously considered by the authors. In the case $k=2$, we show that idempotent generation of the semigroup for all choices of $a$ is equivalent to a condition on the permutation group $G$, stronger than primitivity, which we call the \emph{road closure condition}. We cannot determine all the primitive groups with this property, but we give a conjecture about their classification, and a body of evidence (both theoretical and computational) in support of the conjecture. The paper ends with some problems.

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arXiv2016

Algebraic properties of chromatic roots

A \emph{chromatic root} is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments. We conjecture that, for every algebraic integer $α$, there is a natural number $n$ such that $α+n$ is a chromatic root. This is proved for quadratic integers, an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

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arXiv2016

Generating sets of finite groups

We investigate the extent to which the exchange relation holds in finite groups $G$. We define a new equivalence relation $\equiv_{\mathrm{m}}$, where two elements are equivalent if each can be substituted for the other in any generating set for $G$. We then refine this to a new sequence $\equiv_{\mathrm{m}}^{(r)}$ of equivalence relations by saying that $x \equiv_{\mathrm{m}}^{(r)}y$ if each can be substituted for the other in any $r$-element generating set. The relations $\equiv_{\mathrm{m}}^{(r)}$ become finer as $r$ increases, and we define a new group invariant $ψ(G)$ to be the value of $r$ at which they stabilise to $\equiv_{\mathrm{m}}$. Remarkably, we are able to prove that if $G$ is soluble then $ψ(G) \in \{d(G), d(G) +1\}$, where $d(G)$ is the minimum number of generators of $G$, and to classify the finite soluble groups $G$ for which $ψ(G) = d(G)$. For insoluble $G$, we show that $d(G) \leq ψ(G) \leq d(G) + 5$. However, we know of no examples of groups $G$ for which $ψ(G) > d(G) + 1$. As an application, we look at the generating graph of $G$, whose vertices are the elements of $G$, the edges being the $2$-element generating sets. Our relation $\equiv_{\mathrm{m}}^{(2)}$ enables us to calculate $\mathrm{Aut}(Γ(G))$ for all soluble groups $G$ of nonzero spread, and give detailed structural information about $\mathrm{Aut}(Γ(G))$ in the insoluble case.

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arXiv2016

Triple arrays from difference sets

This paper addresses the question whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having $-1$ as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We also give an alternative and more direct version of the construction, leaving out the intermediate step via Youden squares. This is used when we analyse the case of non-abelian difference sets, for which we prove a sufficient condition for giving triple arrays. We do a computer search for such non-abelian difference sets, but have not found any examples satisfying the given condition.

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arXiv2016

Highest rank of a polytope for $A_n$

We prove that the highest rank of a string C-group constructed from an alternating group $Alt_n$ is 0 if $n=3, 4, 6, 7, 8$; 3 if $n=5$; 4 if $n=9$; 5 if $n=10$; 6 if $n=11$; and $\lfloor\frac{n-1}{2}\rfloor$ if $n\geq 12$. This solves a conjecture made by the last three authors in 2012.

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arXiv2016

$Z_4$-codes and their Gray map images as orthogonal arrays

A classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter. We show that Delsarte's observation extends to codes over arbitrary finite rings. Since the paper of Hammons \emph{et al.}, there is a lot of interest in codes over rings, especially in codes over $Z_4$ and their (usually non-linear) binary Gray map images. We show that Delsarte's observation extends to codes over arbitrary finite commutative rings with identity. Also, we show that the strength of the Gray map image of a $Z_4$ code is one less than the minimum Lee weight of its Gray map image.

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arXiv2016

On the structure of the power graph and the enhanced power graph of a group

Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the \emph{commuting graph} by introducing a new graph which lies in between. We call this new graph as the \emph{enhanced power graph}. For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal.

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arXiv2015

Orbits of Primitive $k$-Homogenous Groups on $(n-k)$-Partitions with Applications to Semigroups

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); in particular we show that $2$-homogeneous groups are $2$-generated. We also describe the orbits of $k$-homogenous groups on partitions with $n-k$ parts, classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partitions are invariant under the normalizer of $G$ in $S_n$, and describe the normalizers of $2$-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

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arXiv2015

Between primitive and $2$-transitive: Synchronization and its friends

An automaton is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing $n$-state automaton has a reset word of length at most $(n-1)^2$. The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group $G$ on a set $Ω$ is said to synchronize a map $f$ if the monoid $\langle G,f\rangle$ generated by $G$ and $f$ is synchronizing in the above sense; we say $G$ is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and $2$-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture, some challenges to finite geometers, some thoughts about infinite analogues, and a long list of open problems.

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arXiv2015

Guessing Games on Triangle-free Graphs

The guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markstrom introduced a method to bound the value of the guessing number from below using the fractional clique number Kf(G). In particular they showed gn(G) >= |V(G)| - Kf(G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50.

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arXiv2015

Asymmetric Latin squares, Steiner triple systems, and edge-parallelisms

This article, showing that almost all objects in the title are asymmetric, is re-typed from a manuscript I wrote somewhere around 1980 (after the papers of Bang and Friedland on the permanent conjecture but before those of Egorychev and Falikman). I am not sure of the exact date. The manuscript had been lost, but surfaced among my papers recently. I am grateful to Laci Babai and Ian Wanless who have encouraged me to make this document public, and to Ian for spotting a couple of typos. In the section on Latin squares, Ian objects to my use of the term "cell"; this might be more reasonably called a "triple" (since it specifies a row, column and symbol), but I have decided to keep the terminology I originally used. The result for Latin squares is in B. D. McKay and I. M. Wanless, On the number of Latin squares, Annals of Combinatorics 9 (2005), 335-344 (arXiv 0909.2101), while the result for Steiner triple systems is in L. Babai, Almost all Steiner triple systems are asymmetric, Annals of Discrete Mathematics 7 (1980), 37-39.

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arXiv2015

Primitive groups and synchronization

Let $Ω$ be a set of cardinality $n$, $G$ a permutation group on $Ω$, and $f:Ω\toΩ$ a map which is not a permutation. We say that $G$ \emph{synchronizes} $f$ if the transformation semigroup $\langle G,f\rangle$ contains a constant map, and that $G$ is a \emph{synchronizing group} if $G$ synchronizes \emph{every} non-permutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it has previously been conjectured that a primitive group synchronizes every non-uniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks $5$ and $6$. In addition we produce graphs whose automorphism groups have approximately $\sqrt{n}$ \emph{non-synchronizing ranks}, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree $n$ synchronizes every non-uniform transformation of rank $n-1$ and $n-2$, and here this is extended to $n-3$ and $n-4$. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.

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arXiv2015

Constructing flag-transitive, point-imprimitive designs

We give a construction of a family of designs with a specified point-partition, and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric $2$-$(1408,336,80)$ design with automorphism group $2^{12}:((3\cdot\mathrm{M}_{22}):2)$, and a construction of one of the families of the symplectic designs (the designs $S^-(n)$) exhibiting a flag-transitive, point-imprimitive automorphism group.

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arXiv2015

Most switching classes with primitive automorphism groups contain graphs with trivial groups

The operation of switching a graph $Γ$ with respect to a subset $X$ of the vertex set interchanges edges and non-edges between $X$ and its complement, leaving the rest of the graph unchanged. This is an equivalence relation on the set of graphs on a given vertex set, so we can talk about the automorphism group of a switching class of graphs. It might be thought that switching classes with many automorphisms would have the property that all their graphs also have many automorphisms. However the main theorem of this paper shows a different picture: with finitely many exceptions, if a non-trivial switching class $\mathcal{S}$ has primitive automorphism group, then it contains a graph whose automorphism group is trivial. We also find all the exceptional switching classes; up to complementation, there are just six.

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arXiv2015

Chains of subsemigroups

We investigate the maximum length of a chain of subsemigroups in various classes of semigroups, such as the full transformation semigroups, the general linear semigroups, and the semigroups of order-preserving transformations of finite chains. In some cases, we give lower bounds for the total number of subsemigroups of these semigroups. We give general results for finite completely regular and finite inverse semigroups. Wherever possible, we state our results in the greatest generality; in particular, we include infinite semigroups where the result is true for these. The length of a subgroup chain in a group is bounded by the logarithm of the group order. This fails for semigroups, but it is perhaps surprising that there is a lower bound for the length of a subsemigroup chain in the full transformation semigroup which is a constant multiple of the semigroup order.

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arXiv2014

String C-groups as transitive subgroups of Sym(n)

If $Γ$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sym(n) (other than Sym(n) and the alternating group Alt(n)), then the rank of $Γ$ is at most $n/2+1$, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.

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arXiv2014

On optimality and construction of circular repeated-measurements designs

The aim of this paper is to characterize and construct universally optimal designs among the class of circular repeated-measurements designs when the parameters do not permit balance for carry-over effects. It is shown that some circular weakly neighbour balanced designs defined by Filipiak and Markiewicz These results extend the work of Magda, Kunert, Filipiak and Markiewicz.

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