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Annales de l’Institut Henri Poincaré D

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Volume 8, Issue 2, 2021, pp. 179–200
DOI: 10.4171/AIHPD/100

Published online: 2021-03-03

Mixed partition functions and exponentially bounded edge-connection rank

Guus Regts[1] and Bart Sevenster[2]

(1) University of Amsterdam, The Netherlands
(2) University of Amsterdam, The Netherlands

We study graph parameters whose associated edge-connection matrices have exponentially bounded rank growth. Our main result is an explicit construction of a large class of graph parameters with this property that we call mixed partition functions. Mixed partition functions can be seen as a generalization of partition functions of vertex models, as introduced by de la Harpe and Jones, [P. de la Harpe and V. F. R. Jones, Graph invariants related to statistical mechanical models: examples and problems, J. Combin. Theory Ser. B 57 (1993), no. 2, 207–227.] and they are related to invariant theory of orthosymplectic supergroup. We moreover show that evaluations of the characteristic polynomial of a simple graph are examples of mixed partition functions, answering a question of de la Harpe and Jones. (NOTE. Some of the results of this paper were announced in an extended abstract: G. Regts and B. Sevenster, Partition functions from orthogonal and symplectic group invariants, Electron. Notes Discrete Math. 61 (2017), 1011–1017. Unfortunately that reference contains a mistake; we will comment on that below).

Keywords: Partition function, graph parameter, orthogonal group, symplectic group, orthosymplectic Lie super algebra, circuit partition polynomial, connection matrix

Regts Guus, Sevenster Bart: Mixed partition functions and exponentially bounded edge-connection rank. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), 179-200. doi: 10.4171/AIHPD/100