Primitive and Imprimitive Graphs

Smith, Derek H.

doi:10.1093/qmath/22.4.551

Cited by 80 publications

(47 citation statements)

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“…On the one hand, we have the imprimitive ones: these have a nontrivial equivalence relation definable without parameters (equivalently, invariant under the full automorphism group). In the finite distance transitive case, these are described by Smith's Theorem [Smi71]; this theorem is an effective tool for reduction to the primitive case when the graph is finite, and also to a significant degree when the graph is infinite.…”

Section: Metrically Homogeneous Graphs: Toward a Catalogmentioning

confidence: 99%

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Two problems on homogeneous structures, revisited

Cherlin¹

2011

Contemporary Mathematics

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Abstract. We take up Peter Cameron's problem of the classification of countably infinite graphs which are homogeneous as metric spaces in the graph metric [Cam98], working toward an explicit catalog of "known" examples on the one hand, and an investigation of the cases which occur as exceptional cases from the perspective of the catalog.We begin with a presentation of Fraïssé's theory of amalgamation classes and the classification of homogeneous structures, with emphasis on the case of homogeneous metric spaces, from the discovery of the Urysohn space to the connection with topological dynamics developed in [KPT05]. We then turn to a discussion of the case of metrically homogeneous graphs.We also take this opportunity to revisit another old chestnut from the theory of homogeneous structures, namely the problem of approximating the generic triangle free graph by finite graphs. Very little is known about this, but it is possible to rephrase the problem in somewhat more explicit geometric terms. And in that form one can raise questions that seem appropriate for design theorists, as well as some questions that involve structures small enough to be explored computationally.

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Section: Metrically Homogeneous Graphs: Toward a Catalogmentioning

confidence: 99%

“…We now turn to Smith's Theorem, a general description of the imprimitive case, following [AH06] (cf. [BCN89,Smi71]). This result applies to imprimitive distance transitive graphs (that is, the homogeneity condition is assumed to hold for pairs of vertices), and even more generally in the finite case.…”

Section: Proof We Consider a Two-point Amalgam Withmentioning

confidence: 99%

Two problems on homogeneous structures, revisited

Cherlin¹

2011

Contemporary Mathematics

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“…Beginning from the seminal paper [99] and the background text [36], antipodal DRGs are playing a special role in the general theory. As we mentioned already earlier, together with [10], the thesis [66] remains the most significant comprehensive source of information about such graphs.…”

Section: Antipodal Coversmentioning

confidence: 99%

“…A census of his collection was published in [33]. In [99] the Foster graph was recognized as a 3-fold antipodal cover of the incidence graph of W 2 . An explicit construction of the incidence structure W 2 , corresponding to the Foster graph, was given in [57].…”

Section: The Foster Graph and Tilde Geometriesmentioning

confidence: 99%

A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices

Klin

Pech²

2011

Ars Math. Contemp.

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New constructions of regular distance regular antipodal covers (in the sense of GodsilHensel) of complete graphs K n are presented. The main source of these constructions are skew generalized Hadamard matrices. It is described how to produce such a matrix of order n 2 over a group T from an arbitrary given generalized Hadamard matrix of order n over the same group T . Further, a new regular cover of K 45 on 135 vertices is produced with the aid of a decoration of the alternating group A 6 .

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“…This partition is not the antipodal partition , since M is transitive on À ( 2 ) and hence , by [23] , it must be the bipartition ͕ ⌬…”

Section: T He a Lmost S Imple C Asementioning

confidence: 99%

Antipodal Distance-transitive Covers of Complete Bipartite Graphs

Иванов

Liebler

Penttila

et al. 1997

European Journal of Combinatorics

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This paper completes the classification of antipodal distance-transitive covers of the complete bipartite graphs K k , k , where k у 3 . For such a cover the antipodal blocks must have size r р k .Although the case r ϭ k has already been considered , we give a unified treatment of r р k . We use deep group-theoretic results as well as representation-theoretic data about explicit linear groups and group coset geometries .Apart from the generic examples arising from finite projective spaces , there are three sporadic examples (arising from the outer automorphisms of the symmetric group S 6 and of the Mathieu group M 1 2 and one related to non-abelian Singer groups on PG 2 (4)) and an infinite family having solvable automorphism group (and with parameters r ϭ q b , k ϭ q a , whereand q is a prime power) . Smith [23] showed that there were only two types of partitions À which can be preserved by Aut G , for a finite imprimitive distance-transitive graph ⌫ of valency at least 3 , namely : (1) À ϭ ͕ ⌬ 1 , ⌬ 2 ͖ , and each edge of ⌫ joins a vertex of ⌬ 1 to a vertex of ⌬ 2 , in which case ⌫ is bipartite ; andHere we use the notationV , which is then called the antipodal partition of V . With any antipodal graph ⌫ we can associate a natural quotient graph , which we shall denote by ⌫ Ј , the vertex set of which is the antipodal partition of ⌫ , two antipodal blocks being adjacent in ⌫ Ј whenever they contain adjacent vertices of ⌫ . If ⌫ is antipodal and its antipodal blocks have size r , then ⌫ is called an r -fold antipodal co er of ⌫ Ј . Smith [23] showed , moreover , that if ⌫ is a finite antipodal distance-transitive graph then its antipodal quotient ⌫ Ј is also distance-transitive . Thus , as part of the problem of classifying finite distance-transitive graphs , all distance-transitive , antipodal covers of the known distance-transitive graphs must be determined . This has been done by J . van

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Primitive and Imprimitive Graphs

Cited by 80 publications

References 0 publications

Two problems on homogeneous structures, revisited

Two problems on homogeneous structures, revisited

A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices

Antipodal Distance-transitive Covers of Complete Bipartite Graphs

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