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Entrywise Logarithmic Matrix Algebra and Dichotomy of Planar Graph Homomorphisms (Part I)
We prove a complexity classification of counting planar graph homomorphisms with non-negative weights. For a real symmetric matrix $M$ with non-negative entries, the problem $\PlGH(M)$ is either (1) P-time computable over all graphs, or (2) \#P-hard in general but P-time computable over planar graph
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