Séminaire ART
organisé par l'équipe Algèbre, représentations, topologie
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Andrew Goodall
Graph invariants from counting homomorphisms to Cayley graphs on Abelian groups
26 mai 2026 - 14:00Salle de séminaires IRMA
The number of homomorphisms from a graph G to graphs G_q indexed by a positive integer q defines an invariant of G in the parameter q. We shall take G_q to be a Cayley graph on an Abelian group. (For the cognoscenti: because we are then effectively counting tensions and can then define a dual invariant counting the corresponding flows.) Of particular interest have been those sequences for which a polynomial in q results, not least because you can evaluate a polynomial invariant in q at values of q that are not positive integers and sometimes make sense of what this invariant says about G. The classical case is where (G_q) is the sequence of complete graphs on q vertices, for which we obtain the chromatic polynomial of G, and if for instance I evaluate the chromatic polynomial at q = -1, then I get the number of acyclic orientations of G. In this talk I will try to explain why we might wish to cast our net further out and seek those sequences (G_q) for which the homomorphism counts from G have a rational generating function (which includes the case where the counts are polynomial in q). To help do so, I will use two examples, the first where the Abelian group on which G_q is defined is cyclic order q, and the second where the Abelian group is a q-fold direct sum of the group of order 2. Joint work with Delia Garijo (Univ. Seville) and Anna de Mier (UPC Barcelona) -
Christian Kassel
tba
2 juin 2026 - 14:00Salle de séminaires IRMA