Graph parameters from symplectic group invariants

TL;DR

This paper introduces a new class of graph parameters derived from tensor invariants of the symplectic group, expanding the landscape of graph invariants beyond existing models and linking them to evaluations of the cycle partition polynomial.

Contribution

It characterizes a novel class of graph parameters from symplectic group invariants, distinct from traditional vertex model partition functions, and connects them to cycle partition polynomial evaluations.

Findings

New class of graph parameters from symplectic tensor invariants

Connections to cycle partition polynomial evaluations

Distinct from existing vertex model invariants

Abstract

In this paper we introduce, and characterize, a class of graph parameters obtained from tensor invariants of the symplectic group. These parameters are similar to 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]. Yet they give a completely different class of graph invariants. We moreover show that certain evaluations of the cycle partition polynomial, as defined by Martin [P. Martin, Enum\'erations eul\'eriennes dans les multigraphes et invariants de Tutte-Grothendieck, Diss. Institut National Polytechnique de Grenoble-INPG; Universit\'e Joseph-Fourier-Grenoble I, 1977], give examples of graph parameters that can be obtained this way.