ORCID Profile
0000-0003-3130-9505
Current Organisation
University of St Andrews
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Combinatorics and Discrete Mathematics (excl. Physical Combinatorics) | Analysis of Algorithms and Complexity | Pure Mathematics | Algebra and Number Theory
Expanding Knowledge in the Information and Computing Sciences | Expanding Knowledge in the Mathematical Sciences |
Publisher: Oxford University Press
Date: 18-01-2007
Publisher: Springer Netherlands
Date: 1977
Publisher: Elsevier BV
Date: 12-2005
Publisher: Springer Science and Business Media LLC
Date: 09-1988
DOI: 10.1007/BF02126798
Publisher: Springer Science and Business Media LLC
Date: 03-1974
DOI: 10.1007/BF01189254
Publisher: Elsevier BV
Date: 09-1992
Publisher: Informa UK Limited
Date: 2015
Publisher: Elsevier BV
Date: 02-2010
Publisher: Informa UK Limited
Date: 13-09-2022
Publisher: Elsevier BV
Date: 03-1973
Publisher: Elsevier BV
Date: 10-2003
Publisher: Elsevier BV
Date: 12-2006
Publisher: Elsevier BV
Date: 02-2022
Publisher: Elsevier BV
Date: 1978
Publisher: Wiley
Date: 1993
DOI: 10.1112/BLMS/25.1.1
Publisher: Wiley
Date: 11-1989
Publisher: Springer US
Date: 2001
Publisher: Wiley
Date: 1995
Publisher: Elsevier BV
Date: 03-2007
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 08-02-2017
DOI: 10.4171/PM/2003
Publisher: Elsevier BV
Date: 10-1979
Publisher: Elsevier BV
Date: 03-1994
Publisher: Elsevier BV
Date: 04-2016
Publisher: Elsevier BV
Date: 1991
Publisher: Elsevier BV
Date: 10-2000
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 08-02-2018
DOI: 10.4171/PM/2000
Publisher: Wiley
Date: 1980
Publisher: Elsevier BV
Date: 11-1976
Publisher: Cambridge University Press (CUP)
Date: 11-1978
DOI: 10.1017/S1446788700011824
Abstract: Groups with the property of the title were considered by Chillag (1977) this paper completes his results by showing that, with known exceptions, they are triply transitive.
Publisher: Springer Science and Business Media LLC
Date: 1988
DOI: 10.1007/BF00191931
Publisher: Informa UK Limited
Date: 21-09-2007
Publisher: Elsevier BV
Date: 12-2002
Publisher: Cambridge University Press (CUP)
Date: 09-1993
DOI: 10.1017/S0963548300000651
Abstract: The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more group-theoretic flavour. Two ex les considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions).
Publisher: Wiley
Date: 04-1981
Publisher: Elsevier BV
Date: 03-1989
Publisher: Cambridge University Press (CUP)
Date: 03-1998
DOI: 10.1017/S096354839700312X
Abstract: We show that if S 1 is a strongly complete sum-free set of positive integers, and if S 0 is a finite sum-free set, then, with positive probability, a random sum-free set U contains S 0 and is contained in S 0 ∪ S 1 . As a corollary we show that, with positive probability, 2 is the only even element of a random sum-free set.
Publisher: Elsevier BV
Date: 12-1978
Publisher: American Mathematical Society (AMS)
Date: 04-04-2018
DOI: 10.1090/TRAN/7248
Abstract: We investigate the extent to which the exchange relation holds in finite groups G G . We define a new equivalence relation ≡ m \\equiv _{\\mathrm {m}} , where two elements are equivalent if each can be substituted for the other in any generating set for G G . We then refine this to a new sequence ≡ m ( r ) \\equiv _{\\mathrm {m}}^{(r)} of equivalence relations by saying that x ≡ m ( r ) y x \\equiv _{\\mathrm {m}}^{(r)}y if each can be substituted for the other in any r r -element generating set. The relations ≡ m ( r ) \\equiv _{\\mathrm {m}}^{(r)} become finer as r r increases, and we define a new group invariant ψ ( G ) \\psi (G) to be the value of r r at which they stabilise to ≡ m \\equiv _{\\mathrm {m}} . Remarkably, we are able to prove that if G G is soluble, then ψ ( G ) ∈ { d ( G ) , \\psi (G) \\in \\{d(G), d ( G ) + 1 } d(G) +1\\} , where d ( G ) d(G) is the minimum number of generators of G G , and to classify the finite soluble groups G G for which ψ ( G ) = d ( G ) \\psi (G) = d(G) . For insoluble G G , we show that d ( G ) ≤ ψ ( G ) ≤ d ( G ) + 5 d(G) \\leq \\psi (G) \\leq d(G) + 5 . However, we know of no ex les of groups G G for which ψ ( G ) d ( G ) + 1 \\psi (G) d(G) + 1 . As an application, we look at the generating graph Γ ( G ) \\Gamma (G) of G G , whose vertices are the elements of G G , the edges being the 2 2 -element generating sets. Our relation ≡ m ( 2 ) \\equiv _{\\mathrm {m}}^{(2)} enables us to calculate A u t ( Γ ( G ) ) \\mathrm {Aut}(\\Gamma (G)) for all soluble groups G G of nonzero spread and to give detailed structural information about A u t ( Γ ( G ) ) \\mathrm {Aut}(\\Gamma (G)) in the insoluble case.
Publisher: Springer Science and Business Media LLC
Date: 10-1973
DOI: 10.1007/BF00147860
Publisher: Statistica Sinica (Institute of Statistical Science)
Date: 2017
Publisher: Elsevier BV
Date: 11-2001
Publisher: Elsevier BV
Date: 12-1991
Publisher: Elsevier BV
Date: 12-2023
Publisher: Elsevier BV
Date: 08-1993
Publisher: Wiley
Date: 07-1974
DOI: 10.1112/BLMS/6.2.136
Publisher: Elsevier BV
Date: 08-2012
Publisher: Springer Science and Business Media LLC
Date: 12-1981
DOI: 10.1007/BF02579455
Publisher: Springer Science and Business Media LLC
Date: 04-2011
Publisher: Springer Science and Business Media LLC
Date: 09-09-2019
Publisher: Wiley
Date: 09-1983
Publisher: Elsevier BV
Date: 07-2011
Publisher: Springer Netherlands
Date: 1997
Publisher: Springer Science and Business Media LLC
Date: 04-07-2022
DOI: 10.1007/S10623-021-00907-2
Abstract: In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\\ge 2$$ m ≥ 2 , a set of $$m+1$$ m + 1 partitions of a set $$\\Omega $$ Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $$m=2$$ m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G . In this paper we investigate what happens if we have $$m+r$$ m + r partitions with $$r\\ge 2$$ r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If $$m=2$$ m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an ex le to show the groups need not be all isomorphic. For $$m $$ m 2 , things are more restricted. Any $$m+1$$ m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G . It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p , then the structure corresponds exactly to an arc of cardinality $$m+r$$ m + r in the $$(m-1)$$ ( m - 1 ) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give ex les where G is the elementary abelian group of order q . These ex les can be lifted to non-elementary abelian groups using p -adic techniques.
Publisher: Elsevier BV
Date: 12-1981
Publisher: Elsevier BV
Date: 10-2023
Publisher: Elsevier BV
Date: 06-2022
Publisher: Elsevier BV
Date: 07-1987
Publisher: Oxford University Press (OUP)
Date: 1987
Publisher: Springer Science and Business Media LLC
Date: 05-1995
DOI: 10.1007/BF01266318
Publisher: Springer Science and Business Media LLC
Date: 31-03-2020
DOI: 10.1007/S00373-020-02162-Z
Abstract: Let G be a group. The power graph of G is a graph with vertex set G in which two distinct elements x , y are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph on the set $$G-\\{1\\}$$ G - { 1 } . A group whose proper power graph is connected must be either a torsion group or a torsion-free group we give characterizations of some groups whose proper power graphs are connected.
Publisher: Elsevier BV
Date: 08-2023
Publisher: Elsevier BV
Date: 10-1998
Publisher: Springer Science and Business Media LLC
Date: 21-07-2007
Publisher: SAGE Publications, Inc.
Date: 2022
Publisher: Springer Berlin Heidelberg
Date: 1997
Publisher: Wiley
Date: 08-1976
Publisher: Elsevier BV
Date: 1975
Publisher: Mathematical Society of Japan (Project Euclid)
Date: 10-1973
Publisher: Elsevier BV
Date: 09-2017
Publisher: Elsevier BV
Date: 05-1988
Publisher: Cambridge University Press
Date: 27-12-1979
Publisher: Wiley
Date: 03-10-2016
DOI: 10.1112/PLMS/PDW040
Publisher: Elsevier BV
Date: 03-2005
Publisher: IOP Publishing
Date: 06-2006
Publisher: Wiley
Date: 03-12-2018
DOI: 10.1002/JCD.21634
Publisher: Elsevier BV
Date: 12-1991
Publisher: Elsevier BV
Date: 03-2005
Publisher: Springer Science and Business Media LLC
Date: 12-1987
DOI: 10.1007/BF00960072
Publisher: Springer Science and Business Media LLC
Date: 29-10-2019
Publisher: Springer Science and Business Media LLC
Date: 02-1994
DOI: 10.1007/BF01198674
Publisher: Wiley
Date: 11-1969
DOI: 10.1112/BLMS/1.3.349
Publisher: Springer Science and Business Media LLC
Date: 02-07-2022
Publisher: Elsevier BV
Date: 03-2008
Publisher: Informa UK Limited
Date: 12-01-2009
Publisher: Springer Science and Business Media LLC
Date: 13-03-2022
DOI: 10.1007/S00026-022-00576-5
Abstract: The power graph P ( G ) of a finite group G is the undirected simple graph with vertex set G , where two elements are adjacent if one is a power of the other. In this paper, the matching numbers of power graphs of finite groups are investigated. We give upper and lower bounds, and conditions for the power graph of a group to possess a perfect matching. We give a formula for the matching number for any finite nilpotent group. In addition, using some elementary number theory, we show that the matching number of the enhanced power graph $$P_e(G)$$ P e ( G ) of G (in which two elements are adjacent if both are powers of a common element) is equal to that of the power graph of G .
Publisher: Elsevier BV
Date: 11-1987
Publisher: Wiley
Date: 20-07-2005
Publisher: Cambridge University Press
Date: 04-02-1999
Abstract: Permutation groups are one of the oldest topics in algebra. However, their study has recently been revolutionised by new developments, particularly the classification of finite simple groups, but also relations with logic and combinatorics, and importantly, computer algebra systems have been introduced that can deal with large permutation groups. This book gives a summary of these developments, including an introduction to relevant computer algebra systems, sketch proofs of major theorems, and many ex les of applying the classification of finite simple groups. It is aimed at beginning graduate students and experts in other areas, and grew from a short course at the EIDMA institute in Eindhoven.
Publisher: Informa UK Limited
Date: 05-01-2003
Publisher: Princeton University Press
Date: 31-12-2010
Publisher: The Electronic Journal of Combinatorics
Date: 02-07-2021
DOI: 10.37236/9961
Abstract: The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.
Publisher: Oxford University Press (OUP)
Date: 1975
Publisher: WORLD SCIENTIFIC
Date: 07-2009
Publisher: Elsevier BV
Date: 09-1976
Publisher: Elsevier BV
Date: 12-2006
Publisher: Springer Science and Business Media LLC
Date: 06-1973
DOI: 10.1007/BF00184629
Publisher: Springer Science and Business Media LLC
Date: 09-1992
DOI: 10.1007/BF00151520
Publisher: Elsevier BV
Date: 02-1994
Publisher: Cambridge University Press
Date: 10-06-1976
Publisher: Oxford University Press (OUP)
Date: 1985
Publisher: Elsevier BV
Date: 2010
Publisher: Elsevier BV
Date: 12-2006
Publisher: American Mathematical Society (AMS)
Date: 03-05-2018
DOI: 10.1090/TRAN/7274
Publisher: Wiley
Date: 08-08-2022
DOI: 10.1002/JGT.22871
Abstract: The purpose of this note is to define a graph whose vertex set is a finite group , whose edge set is contained in that of the commuting graph of and contains the enhanced power graph of . We call this graph the deep commuting graph of . Two elements of are joined in the deep commuting graph if and only if their inverse images in every central extension of commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of act as automorphisms of the deep commuting graph.
Publisher: Oxford University Press (OUP)
Date: 1983
Publisher: Cambridge University Press (CUP)
Date: 26-10-2006
Publisher: Elsevier BV
Date: 02-2016
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 13-11-2017
DOI: 10.4171/EMSS/4-2-1
Publisher: Elsevier BV
Date: 05-1989
Publisher: Elsevier BV
Date: 10-2019
Publisher: Springer Science and Business Media LLC
Date: 11-1976
DOI: 10.1007/BF00150782
Publisher: Springer Science and Business Media LLC
Date: 1972
DOI: 10.1007/BF01111509
Publisher: Elsevier BV
Date: 03-2020
Publisher: Elsevier BV
Date: 04-1997
Publisher: Wiley
Date: 09-1972
Publisher: Elsevier BV
Date: 04-2005
Publisher: Elsevier BV
Date: 11-1982
Publisher: Elsevier BV
Date: 03-2009
Publisher: Springer Science and Business Media LLC
Date: 05-1974
DOI: 10.1007/BF00181362
Publisher: Springer Science and Business Media LLC
Date: 17-02-2007
Publisher: Elsevier BV
Date: 03-1989
Publisher: Elsevier BV
Date: 2013
Publisher: Cambridge University Press
Date: 22-10-2015
Publisher: Springer Science and Business Media LLC
Date: 1972
DOI: 10.1007/BF01113925
Publisher: Cambridge University Press
Date: 10-06-1976
Abstract: These notes present an investigation of a condition similar to Euclid's parallel axiom for subsets of finite sets. The background material to the theory of parallelisms is introduced and the author then describes the links this theory has with other topics from the whole range of combinatorial theory and permutation groups. These include network flows, perfect codes, Latin squares, block designs and multiply-transitive permutation groups, and long and detailed appendices are provided to serve as introductions to these various subjects. Many of the results are published for the first time.
Publisher: Cambridge University Press (CUP)
Date: 03-1993
DOI: 10.1017/S0963548300000444
Abstract: In this paper we prove that given a finite collection of finite graphs, and the subsets of vertices of a random graph G that induce those graphs, it is almost always possible to uniquely reconstruct a class of graphs equivalent to G .
Publisher: Princeton University Press
Date: 31-12-2010
Publisher: Wiley
Date: 02-1982
Publisher: Springer Science and Business Media LLC
Date: 12-1985
DOI: 10.1007/BF02582937
Publisher: Walter de Gruyter GmbH
Date: 2010
DOI: 10.1515/JGT.2010.023
Publisher: Canadian Mathematical Society
Date: 12-2000
Abstract: A binary structure S has the pigeonhole property ( 𝒫 ) if every finite partition of S induces a block isomorphic to S . We classify all countable tournaments with ( 𝒫 ) the class of orders with ( 𝒫 ) is completely classified.
Publisher: Elsevier BV
Date: 06-1977
Publisher: Elsevier BV
Date: 10-1992
Publisher: Elsevier BV
Date: 05-2003
Publisher: Elsevier BV
Date: 04-1980
Publisher: Elsevier BV
Date: 2021
Publisher: Elsevier BV
Date: 06-1989
Publisher: Theory of Computing Exchange
Date: 2014
Publisher: Elsevier BV
Date: 12-2003
Publisher: Elsevier BV
Date: 11-1995
Publisher: Theory of Computing Exchange
Date: 2014
Publisher: Springer Science and Business Media LLC
Date: 12-1974
DOI: 10.1007/BF01214373
Publisher: Elsevier BV
Date: 06-2013
Publisher: Wiley
Date: 04-1983
Publisher: Wiley
Date: 21-01-2011
DOI: 10.1112/BLMS/BDQ096
Publisher: Cambridge University Press (CUP)
Date: 06-2001
DOI: 10.2307/2695036
Abstract: We can use the compositional semantics of Hodges [9] to show that any compositional semantics for logics of imperfect information must obey certain constraints on the number of semantically inequivalent formulas. As a corollary, there is no compositional semantics for the ‘independence-friendly’ logic of Hintikka and Sandu (henceforth IF) in which the interpretation in a structure A of each 1 -ary formula is a subset of the domain of A (Corollary 6.2 below proves this and more). After a fashion, this rescues a claim of Hintikka and provides the proof which he lacked: … there is no realistic hope of formulating compositional truth-conditions for [sentences of IF], even though I have not given a strict impossibility proof to that effect. (Hintikka [6] page 110ff.) One curious spinoff is that there is a structure of cardinality 6 on which the logic of Hintikka and Sandu gives nearly eight million inequivalent formulas in one free variable (which is more than the population of Finland). We thank the referee for a sensible change of notation, and Joel Berman and Stan Burris for bringing us up to date with the computation of Dedekind's function (see section 4). Our own calculations, utterly trivial by comparison, were done with Maple V. The paper Hodges [9] (cf. [10]) gave a compositional semantics for a language with some devices of imperfect information. The language was complicated, because it allowed imperfect information both at quantifiers and at conjunctions and disjunctions.
Publisher: Springer Science and Business Media LLC
Date: 09-1982
DOI: 10.1007/BF01318900
Publisher: Oxford University Press
Date: 27-06-2013
Publisher: Springer Science and Business Media LLC
Date: 28-02-2018
Publisher: Elsevier BV
Date: 09-1990
Publisher: Wiley
Date: 07-1979
Publisher: Elsevier BV
Date: 06-1986
Publisher: Elsevier BV
Date: 02-2004
Publisher: Cambridge University Press
Publisher: Princeton University Press
Date: 18-07-2010
DOI: 10.2307/J.CTT7SD01
Publisher: Springer Science and Business Media LLC
Date: 12-1993
DOI: 10.1007/BF01303511
Publisher: Springer Science and Business Media LLC
Date: 1990
DOI: 10.1007/BF00147347
Publisher: Springer Science and Business Media LLC
Date: 09-05-2018
Publisher: Elsevier BV
Date: 08-1983
Publisher: Springer Science and Business Media LLC
Date: 03-1982
DOI: 10.1007/BF02579277
Publisher: Elsevier BV
Date: 12-1989
Publisher: Wiley
Date: 2005
DOI: 10.1002/JCD.20057
Publisher: Wiley
Date: 09-1997
Publisher: Elsevier BV
Date: 02-1996
Publisher: The Electronic Journal of Combinatorics
Date: 29-01-2021
DOI: 10.37236/9802
Abstract: For a nilpotent group $G$, let $\\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\\Xi(G)$ has vertex set $G \\setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\\Xi^+(G)$ be the subgraph of $\\Xi(G)$ induced by its non-isolated vertices. We show that if $\\Xi(G)$ has an edge, then $\\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\\Xi(G) = \\Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\\Xi(G)$ in more detail.
Publisher: Wiley
Date: 10-1975
Publisher: Wiley
Date: 12-1972
Publisher: Cambridge University Press
Date: 05-04-2001
Publisher: Wiley
Date: 05-06-2007
DOI: 10.1112/BLMS/BDM034
Publisher: Elsevier BV
Date: 08-2006
Publisher: Springer Science and Business Media LLC
Date: 03-2004
Publisher: American Mathematical Society (AMS)
Date: 03-12-2020
DOI: 10.1090/TRAN/8285
Publisher: Elsevier BV
Date: 06-1991
Publisher: Elsevier BV
Date: 11-1995
Publisher: Chapman and Hall/CRC
Date: 02-11-2006
Publisher: Elsevier BV
Date: 07-2011
Publisher: Elsevier BV
Date: 05-2003
Publisher: Springer Science and Business Media LLC
Date: 30-11-2011
Publisher: Walter de Gruyter GmbH
Date: 06-01-2001
Publisher: Springer Science and Business Media LLC
Date: 05-1996
DOI: 10.1007/BF00130572
Publisher: Springer Berlin Heidelberg
Date: 1982
DOI: 10.1007/BFB0061972
Publisher: Wiley
Date: 28-04-2017
DOI: 10.1112/PLMS.12039
Publisher: Elsevier BV
Date: 06-1980
Publisher: Springer US
Date: 1996
Publisher: Springer Science and Business Media LLC
Date: 18-11-2020
DOI: 10.1007/S00010-020-00763-W
Abstract: It is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either $$x\\in y$$ x ∈ y or $$y\\in x$$ y ∈ x ), we obtain the Erdős–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple it may have loops (if $$x\\in x$$ x ∈ x for some x ) or multiple edges (if $$x\\in y$$ x ∈ y and $$y\\in x$$ y ∈ x for some distinct x , y ). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $$\\aleph _0$$ ℵ 0 -categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $$\\aleph _0$$ ℵ 0 -categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.
Publisher: Elsevier BV
Date: 03-1992
Publisher: Elsevier BV
Date: 02-2010
Publisher: Wiley
Date: 24-07-2006
Publisher: Walter de Gruyter GmbH
Date: 30-01-2008
DOI: 10.1515/JGT.2008.053
Publisher: Elsevier BV
Date: 03-2008
Publisher: Elsevier BV
Date: 05-2014
Publisher: WORLD SCIENTIFIC (EUROPE)
Date: 20-04-2016
Publisher: Elsevier BV
Date: 08-1981
Publisher: Springer Science and Business Media LLC
Date: 23-01-2019
Publisher: Wiley
Date: 1975
Publisher: Elsevier BV
Date: 05-2003
Publisher: Elsevier BV
Date: 05-2014
Publisher: Birkhäuser Basel
Date: 2001
Publisher: Cambridge University Press
Publisher: Springer Science and Business Media LLC
Date: 08-05-2017
Publisher: American Mathematical Society (AMS)
Date: 23-05-2022
DOI: 10.1090/TRAN/8507
Publisher: Elsevier
Date: 1980
Publisher: Elsevier
Date: 1980
Publisher: Springer Science and Business Media LLC
Date: 1982
DOI: 10.1007/BF00147332
Publisher: Elsevier BV
Date: 08-1985
Publisher: Elsevier BV
Date: 04-2002
Publisher: Springer Science and Business Media LLC
Date: 03-1973
DOI: 10.1007/BF01213827
Publisher: Springer Science and Business Media LLC
Date: 10-1984
DOI: 10.1007/BF01196649
Publisher: Elsevier BV
Date: 09-1995
Publisher: Elsevier
Date: 1995
Publisher: Elsevier BV
Date: 08-2022
Publisher: Elsevier BV
Date: 10-1987
Publisher: Elsevier BV
Date: 12-2023
Publisher: Cambridge University Press
Date: 29-06-1990
Abstract: The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. They are precisely those structures which are determined by first-order logical axioms together with the assumption of countability. This book concerns such structures, their substructures and their automorphism groups. A wide range of techniques are used: group theory, combinatorics, Baire category and measure among them. The book arose from lectures given at a research symposium and retains their informal style, whilst including as well many recent results from a variety of sources. It concludes with exercises and unsolved research problems.
Publisher: Elsevier BV
Date: 10-1991
Publisher: Elsevier BV
Date: 02-2010
Publisher: Springer New York
Date: 2013
Publisher: Springer Science and Business Media LLC
Date: 06-1977
DOI: 10.1007/BF01215145
Publisher: Elsevier BV
Date: 02-2010
Publisher: American Mathematical Society
Date: 1979
Publisher: Cambridge University Press
Publisher: Elsevier BV
Date: 05-1994
Publisher: Springer Science and Business Media LLC
Date: 20-05-2016
Publisher: Cambridge University Press
Date: 2004
Publisher: Springer Science and Business Media LLC
Date: 1975
DOI: 10.1007/BF00148772
Publisher: Wiley
Date: 11-1999
DOI: 10.1002/(SICI)1097-0118(199911)32:3<229::AID-JGT2>3.0.CO;2-C
Publisher: Elsevier BV
Date: 2006
Publisher: Elsevier BV
Date: 06-2009
Publisher: Cambridge University Press
Publisher: Cambridge University Press
Publisher: Elsevier BV
Date: 12-1986
Publisher: Elsevier BV
Date: 2010
Publisher: Springer Science and Business Media LLC
Date: 02-04-2016
Publisher: Springer Science and Business Media LLC
Date: 13-12-2011
Publisher: Elsevier BV
Date: 10-2022
Publisher: Cambridge University Press
Date: 16-04-1981
Publisher: Elsevier BV
Date: 08-2000
Publisher: Elsevier BV
Date: 02-1994
Publisher: Cambridge University Press (CUP)
Date: 10-2008
DOI: 10.1017/S1446788708000815
Abstract: The core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for ex le, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for ex le, about spreads, ovoids and partitions into ovoids in polar spaces).
Publisher: Elsevier BV
Date: 11-2006
Publisher: Wiley
Date: 06-1978
Publisher: Elsevier BV
Date: 08-2013
Publisher: Elsevier BV
Date: 04-2021
Publisher: Elsevier BV
Date: 08-2000
Publisher: Elsevier BV
Date: 05-2021
Publisher: Walter de Gruyter GmbH
Date: 04-1998
Publisher: Springer Science and Business Media LLC
Date: 08-08-2016
Publisher: Springer Science and Business Media LLC
Date: 23-05-2022
DOI: 10.1007/S00373-022-02496-W
Abstract: Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups. We study some properties of graphs in this new hierarchy. In particular, we characterize the groups for which the graphs are complete, and in most cases, we characterize the dominant vertices (those joined to all others). Also, we give some results about universality, perfectness, and clique number.
Publisher: Wiley
Date: 04-1977
Publisher: Springer London
Date: 1998
Publisher: Elsevier BV
Date: 2015
Publisher: Cambridge University Press (CUP)
Date: 1999
DOI: 10.1017/S0963548398003423
Abstract: This paper begins with the observation that half of all graphs containing no induced path of length 3 are disconnected. We generalize this in several directions. First, we give necessary and sufficient conditions (in terms of generating functions) for the probability of connectedness in a suitable class of graphs to tend to a limit strictly between zero and one. Next we give a general framework in which this and related questions can be posed, involving operations on classes of finite structures. Finally, we discuss briefly an algebra associated with such a class of structures, and give a conjecture about its structure.
Publisher: Elsevier BV
Date: 1978
Publisher: Springer Science and Business Media LLC
Date: 06-1976
DOI: 10.1007/BF01214702
Publisher: Wiley
Date: 08-2006
DOI: 10.1002/JCD.20120
Publisher: Elsevier BV
Date: 11-2007
Publisher: Informa UK Limited
Date: 04-05-2021
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 2007
DOI: 10.4171/RMI/499
Publisher: Springer Berlin Heidelberg
Date: 1982
DOI: 10.1007/BFB0062988
Publisher: Elsevier BV
Date: 03-1982
Publisher: Springer New York
Date: 2013
Publisher: Elsevier BV
Date: 07-2009
Publisher: Elsevier BV
Date: 08-1982
Publisher: Elsevier BV
Date: 1973
Publisher: Wiley
Date: 07-08-2017
DOI: 10.1002/JCD.21569
Publisher: Cambridge University Press (CUP)
Date: 1999
DOI: 10.1017/S0963548398003435
Abstract: Our main topic is the number of subsets of [1, n ] which are maximal with respect to some condition such as being sum-free, having no number iding another, etc . We also investigate some related questions.
Publisher: Wiley
Date: 04-1983
Publisher: Elsevier BV
Date: 04-2005
Publisher: Cambridge University Press
Date: 21-06-2017
Abstract: Enumerative combinatorics, in its algebraic and analytic forms, is vital to many areas of mathematics, from model theory to statistical mechanics. This book, which stems from many years' experience of teaching, invites students into the subject and prepares them for more advanced texts. It is suitable as a class text or for in idual study. The author provides proofs for many of the theorems to show the range of techniques available, and uses ex les to link enumerative combinatorics to other areas of study. The main section of the book introduces the key tools of the subject (generating functions and recurrence relations), which are then used to study the most important combinatorial objects, namely subsets, partitions, and permutations of a set. Later chapters deal with more specialised topics, including permanents, SDRs, group actions and the Redfield–Pólya theory of cycle indices, Möbius inversion, the Tutte polynomial, and species.
Publisher: Springer Science and Business Media LLC
Date: 04-03-2020
DOI: 10.1007/S13253-020-00388-1
Abstract: Square lattice designs are often used in trials of new varieties of various agricultural crops. However, there are no square lattice designs for 36 varieties in blocks of size six for four or more replicates. Here, we use three different approaches to construct designs for up to eight replicates. All the designs perform well in terms of giving a low average variance of variety contrasts. Supplementary materials accompanying this paper appear online.
Publisher: American Mathematical Society (AMS)
Date: 07-2015
DOI: 10.1090/TRAN/6368
Abstract: Let X X be a finite set such that | X | = n |X|=n and let i ⩽ j ⩽ n i\leqslant j \leqslant n . A group G ⩽ S n G\leqslant \mathcal {S}_{n} is said to be ( i , j ) (i,j) -homogeneous if for every I , J ⊆ X I,J\subseteq X , such that | I | = i |I|=i and | J | = j |J|=j , there exists g ∈ G g\in G such that I g ⊆ J Ig\subseteq J . (Clearly ( i , i ) (i,i) -homogeneity is i i -homogeneity in the usual sense.) A group G ⩽ S n G\leqslant \mathcal {S}_{n} is said to have the k k -universal transversal property if given any set I ⊆ X I\subseteq X (with | I | = k |I|=k ) and any partition P P of X X into k k blocks, there exists g ∈ G g\in G such that I g Ig is a section for P P . (That is, the orbit of each k k -subset of X X contains a section for each k k -partition of X X .) In this paper we classify the groups with the k k -universal transversal property (with the exception of two classes of 2 2 -homogeneous groups) and the ( k − 1 , k ) (k-1,k) -homogeneous groups (for 2 k ⩽ ⌊ n + 1 2 ⌋ 2 k\leqslant \lfloor \frac {n+1}{2}\rfloor ). As a corollary of the classification we prove that a ( k − 1 , k ) (k-1,k) -homogeneous group is also ( k − 2 , k − 1 ) (k-2,k-1) -homogeneous, with two exceptions and similarly, but with no exceptions, groups having the k k -universal transversal property have the ( k − 1 ) (k-1) -universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank k k transformation on X X generate a regular semigroup (for 1 ⩽ k ⩽ ⌊ n + 1 2 ⌋ 1\leqslant k\leqslant \lfloor \frac {n+1}{2}\rfloor ). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.
Publisher: Cambridge University Press (CUP)
Date: 03-01-2006
Publisher: Elsevier BV
Date: 09-1992
Publisher: Wiley
Date: 09-1982
Publisher: Michigan Mathematical Journal
Date: 05-2009
Publisher: Cambridge University Press (CUP)
Date: 09-1987
DOI: 10.1017/S0305004100067256
Abstract: A group G is defined to be a B-group if any primitive permutation group which contains G as a regular subgroup is doubly transitive. In the case where G is finite the existence of families of B-groups has been established by Burnside, Schur, Wielandt and others and led to the investigation of S-rings. A survey of this work is given in [3], sections 13·7–13·12. In this paper the possibility of the existence of countable B-groups is discussed. Three distinct methods are given to embed a countable group as a regular subgroup of a simply primitive permutation group, and in each case a condition on the square root sets of elements of the group is necessary for the embedding to be carried out. It is easy to demonstrate that this condition is not sufficient, and the general question remains open.
Publisher: Wiley
Date: 10-2002
Location: United Kingdom of Great Britain and Northern Ireland
Location: United Kingdom of Great Britain and Northern Ireland
Location: United Kingdom of Great Britain and Northern Ireland
Location: United Kingdom of Great Britain and Northern Ireland
Start Date: 05-2011
End Date: 12-2014
Amount: $270,000.00
Funder: Australian Research Council
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