Mixed partition functions and exponentially bounded edge-connection rank

TL;DR

This paper introduces mixed partition functions, a broad class of graph parameters with exponentially bounded edge-connection matrix rank growth, linking them to vertex models, invariant theory, and characteristic polynomial evaluations.

Contribution

It explicitly constructs mixed partition functions, generalizes vertex model partition functions, and connects them to invariant theory and characteristic polynomial evaluations.

Findings

Mixed partition functions have exponentially bounded rank growth.

Evaluations of the characteristic polynomial are examples of mixed partition functions.

The work links mixed partition functions to invariant theory of orthosymplectic supergroup.

Abstract

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, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 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.