Editorial: Special issue on finite geometries in honor of Frank De Clerck

Bamberg, John; Beule, Jan De; Durante, Nicola; Lavrauw, Michel

doi:10.1007/s10623-014-9931-y

Cited by 6 publications

(23 citation statements)

References 36 publications

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“…Γ(u) u = S n for all u ∈ V Γ, in which case the claim holds with G = Aut(Γ). We obtain the same result when n = 5 and Γ is the Conway-Smith graph by [5,Theorem 13.2.3], or when n = 5 and Γ is the commuting involutions graph of the conjugacy class of the involutory Galois field automorphism in PΣL 2 (25) by [5, Proposition 12.2.2], or when n = 6 and Γ is the complement of an elliptic quadric in the graph Sp 6 (2) by GAP [1,10,30]. Lastly, if n = 6 and Γ is the complement of a hyperplane in the graph Sp 6 (2), then Aut(Γ) Γ(u) u Aut(Γ) u = S 6 for all u ∈ V Γ by GAP [1,10,30], in which case the claim holds with G = Aut(Γ).…”

mentioning

confidence: 81%

“…If Γ is connected but not bipartite, then Γ 2 is connected, and if Γ is connected and bipartite, then Γ 2 has exactly two connected components; these are called the halved graphs of Γ. We write 1 2 Γ for a halved graph of Γ whenever the halved graphs of Γ are isomorphic.…”

Section: Preliminariesmentioning

confidence: 99%

“…In particular, C E n , and we may assume that the halved graph of Γ(C) has vertex set E n /C. Note that if C is not an even code, then there is a natural graph isomorphism between 1 2 Γ(C ∩ E n ) and the distance 2 graph Γ(C) 2 . The other basic code is the repetition code {0, 1}, whose coset graph n is called the folded n-cube.…”

Section: Covering Mapsmentioning

confidence: 99%

“…This graph has 36 vertices and diameter 2. Using GAP [1,10,30], we determined that it is distance-transitive with the following distance distribution diagram. We also determined that it has automorphism group PΩ − 6 (2) 2 and vertex stabiliser S 6 × C 2 , where C 2 fixes the neighbourhood of the vertex pointwise.…”

Section: Covering Mapsmentioning

confidence: 99%

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Locally triangular graphs and rectagraphs with symmetry

Bamberg

Devillers

Fawcett

et al. 2015

Journal of Combinatorial Theory, Series A

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Abstract. Locally triangular graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-arc lies in a unique quadrangle. A graph Γ is locally rank 3 if there exists G Aut(Γ) such that for each vertex u, the permutation group induced by the vertex stabiliser Gu on the neighbourhood Γ(u) is transitive of rank 3. One natural place to seek locally rank 3 graphs is among the locally triangular graphs, where every induced neighbourhood graph is isomorphic to a triangular graph Tn. This is because the graph Tn, which has vertex set the 2-subsets of {1, . . . , n} and edge set the pairs of 2-subsets intersecting at one point, admits a rank 3 group of automorphisms. In this paper, we classify the locally 4-homogeneous rectagraphs under some additional structural assumptions. We then use this result to classify the connected locally triangular graphs that are also locally rank 3.

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mentioning

confidence: 81%

Section: Preliminariesmentioning

confidence: 99%

Section: Covering Mapsmentioning

confidence: 99%

Section: Covering Mapsmentioning

confidence: 99%

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Locally triangular graphs and rectagraphs with symmetry

Bamberg

Devillers

Fawcett

et al. 2015

Journal of Combinatorial Theory, Series A

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“…By using the FinInG package [2] for GAP [19], the graph Γ can be constructed as the incidence graph of a block design 1 with the point set {1, 2, . .…”

Section: Distance-regular Line Graphsmentioning

confidence: 99%

Distance-regular graphs admitting a perfect $1$-code

Jazaeri¹

2023

Preprint

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In this paper, we study the problem that which of distance-regular graphs admit a perfect 1-code. Among other results, we characterize distance-regular line graphs which admit a perfect 1-code. Moreover, we characterize all known distanceregular graphs with small valency at most 4, the distance-regular graphs with known putative intersection arrays for valency 5, and all distance-regular graphs with girth 3 and valency 6 or 7 which admit a perfect 1-code. 2020 Mathematics Subject Classification. 05C69 and 05E30.

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Genuinely nonabelian partial difference sets

Polhill,

Davis,

Smith

et al. 2024

J of Combinatorial Designs

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Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

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Editorial: Special issue on finite geometries in honor of Frank De Clerck

Cited by 6 publications

References 36 publications

Locally triangular graphs and rectagraphs with symmetry

Locally triangular graphs and rectagraphs with symmetry

Distance-regular graphs admitting a perfect $1$-code

Genuinely nonabelian partial difference sets

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