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dc.contributor.authorBailey, R. A.
dc.contributor.authorCameron, Peter J.
dc.identifier.citationBailey , R A & Cameron , P J 2021 , ' The diagonal graph ' , Journal of the Ramanujan Mathematical Society , vol. 36 , no. 4 , pp. 353-361 .en
dc.identifier.otherPURE: 274635532
dc.identifier.otherPURE UUID: 1872ac81-870d-4f65-a151-91cf4b200738
dc.identifier.otherORCID: /0000-0003-3130-9505/work/105318437
dc.identifier.otherORCID: /0000-0002-8990-2099/work/105318439
dc.identifier.otherWOS: 000735888500012
dc.identifier.otherScopus: 85129992896
dc.description.abstractAccording to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a much larger class than those occurring in this theorem. For any positive integer m and group G (finite or infinite), there is a diagonal semilattice, a sub-semilattice of the lattice of partitions of a set Ω, whose automorphism group is the corresponding diagonal group. Moreover, there is a graph (the diagonal graph), bearing much the same relation to the diagonal semilattice and group as the Hamming graph does to the Cartesian lattice and the wreath product of symmetric groups. Our purpose here, after a brief introduction to this semilattice and graph, is to establish some properties of this graph. The diagonal graph ΓD(G,m) is a Cayley graph for the group Gm, and so is vertex-transitive. We establish its clique number in general and its chromatic number in most cases, with a conjecture about the chromatic number in the remaining cases. We compute the spectrum of the adjacency matrix of the graph, using a calculation of the Möbius function of the diagonal semilattice. We also compute some other graph parameters and symmetry properties of the graph. We believe that this family of graphs will play a significant role in algebraic graph theory.
dc.relation.ispartofJournal of the Ramanujan Mathematical Societyen
dc.rightsCopyright © 2021 Rumanujan Mathematical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at
dc.subjectQA Mathematicsen
dc.titleThe diagonal graphen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews. Centre for Interdisciplinary Research in Computational Algebraen
dc.contributor.institutionUniversity of St Andrews. Statisticsen
dc.description.statusPeer revieweden

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