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SEMINAR: Groups and Combinatorics Seminar: Cheryl Praeger, 16:00 Friday 04 May 2018 in Weatherburn LT

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Groups and Combinatorics Seminar: Cheryl Praeger, 16:00 Friday 04 May 2018 in Weatherburn LT Other events...
Speaker: Cheryl Praeger (UWA)

Title: Derangement action digraphs and graphs

Time and place: Fri 04 May, 2018 in Weatherburn LT

Abstract: Derangement action digraphs are closely related to group action digraphs, which were introduced by Annexstein, Baumslag and Rosenberg in 1990 as models for interconnection networks underpinning parallel computer architectures. They were introduced by Moharram Iradmusa and me as a generalisation of Cayley digraphs. Some group action digraphs have loops or multiple arcs, and we wished to avoid these. We were in fact looking for a natural family of simple digraphs which properly contains all finite simple vertex-transitive graphs and digraphs, but is not `too large'. We believe that the family of derangement action digraphs satisfies these criteria.

For a non-empty set X and a non-empty subset S of derangements of X (fixed-point-free permutations of X), we define the derangement action digraph DA(X, S) to have vertex set X, and an arc from x to y if and only if y is the image of x under the action of some element of S. Thus by definition it is a simple digraph (no loops, no multiple arcs). It was easy to see that the family of derangement action digraphs contains all Cayley digraphs, but not so easy to see (though it is true) that it also contains all finite vertex-transitive graphs. I’ll report on what we discovered: in particular we found necessary and sufficient conditions on S under which DA(X, S) may be viewed as a simple undirected graph of valency |S|. We investigated various structural and symmetry properties of these digraphs, but are left with several open problems.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html
Contact Stephen Glasby <[email protected]>
Start Fri, 04 May 2018 16:00
End Fri, 04 May 2018 17:00
Submitted by Stephen Glasby <[email protected]>
Last Updated Tue, 01 May 2018 10:53
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