Tutte’s dichromate for signed graphs

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant graphs up to vertex switching that contains among its evaluations number proper colorings and nowhere-zero flows. In this, it parallels polynomial graph, which chromatic flow specializations. The tensions (for they are not simply related for graphs) is given in terms trivariate at two distinct points. Interestingly, bivariate dichromatic biased shown by Zaslavsky share many similar properties with does general yield flows graph. Therefore ``dichromate" (our polynomial) differs from (the rank-size generating function). can be extended ordered pairs matroids on common ground set -- cycle matroid underlying frame form relevant pair matroids. This canonically defined sense recent paper Krajewski, Moffatt Tanasa, was first studied Welsh Kayibi four-variable linking set.