Abstract
AbstractWe develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.
Highlights
Many graph properties can be described in the general framework called graph homomorphisms.Suppose G and H are two graphs
If H = Kq is a clique on q vertices, a vertex map φ : V(G) → {1, . . . , q} is a graph homomorphism iff φ is a proper vertex colouring of G using at most q colours
We show that there is only one condition, which is both necessary and sufficient for a graph parameter to be expressible by graph homomorphisms (GH) with arbitrary vertex and edge weights, and that is a simple exponential bound on the rank of the connection tensor
Summary
Many graph properties can be described in the general framework called graph homomorphisms. Many problems in statistical physics, such as (weighted) orientation problems, ice models, six-vertex models, etc., are all naturally expressible as Holant problems It has been proved [15] that for Holant problems defined by an arbitrary set of complex-valued symmetric constraint functions on Boolean variables, (A) the three-way classification above holds, but (B) holographic reductions to Kasteleyn’s algorithm are not universal for type (2); there is an additional class of planar P-time computable problems; these, together with holographic reductions to Kasteleyn’s algorithm, constitute a complete algorithmic repertoire for this class. . the results in [27] do not answer whether counting perfect matchings, and other similar problems naturally expressible as Holant problems (edge models), can be expressed as partition functions of GH when complex vertex and edge weights are allowed.
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