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

中文翻译：

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有符号图的 Tutte 重铬酸盐

对于有符号图（我们的三变量 Tutte 多项式）与二色多项式（秩大小生成函数）不同。有符号图的三变量 Tutte 多项式可以扩展到公共基础集上有序对阵的不变量——对于有符号图，其基础图的循环拟阵及其框架拟阵形成相关的拟阵对。在 Krajewski、Moffatt 和 Tanasa 最近的一篇论文中，这个不变量是共同基础集上拟阵对的规范定义的 Tutte 多项式，并且首先被威尔士和卡伊比研究为拟阵对的四变量链接多项式一个共同点。有符号图的三变量 Tutte 多项式可以扩展到公共基础集上有序对阵的不变量——对于有符号图，其基础图的循环拟阵及其框架拟阵形成相关的拟阵对。在 Krajewski、Moffatt 和 Tanasa 最近的一篇论文中，这个不变量是共同基础集上拟阵对的规范定义的 Tutte 多项式，并且首先被威尔士和卡伊比研究为拟阵对的四变量链接多项式一个共同点。有符号图的三变量 Tutte 多项式可以扩展到公共基础集上有序对阵的不变量——对于有符号图，其基础图的循环拟阵及其框架拟阵形成相关的拟阵对。在 Krajewski、Moffatt 和 Tanasa 最近的一篇论文中，这个不变量是共同基础集上拟阵对的规范定义的 Tutte 多项式，并且首先被威尔士和卡伊比研究为拟阵对的四变量链接多项式一个共同点。