The Tutte poynomial T(G; x, y) of a graph G is a two-variable graph polynomial, and it gives interesting information about the graph. Many chemically interesting polycyclic polymers can be modeled by uniform or non-uniform polycyclic graphs. In this paper, we consider the Tutte poynomial of several classes of alternating polycyclic chains which contain phenylene chains and their dicyclobutadieno derivatives as special cases. Further, explicit closed formula of the number of spanning trees, the number of spanning forests and the number of spanning connected subgraphs of phenylenes (resp. the dicyclobutadieno derivatives of phenylenes) are obtained.

中文翻译：

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交替多环链的 Tutte 多项式

图 G 的 Tutte 多项式 T(G; x, y) 是一个二变量图多项式，它提供了有关该图的有趣信息。许多化学上有趣的多环聚合物可以通过均匀或非均匀多环图来建模。在本文中，我们将包含亚苯基链及其二环丁二烯衍生物的几类交替多环链的 Tutte 多项式视为特例。进一步得到了生成树数、生成林数和生成亚苯基（和亚苯基的二环丁二烯衍生物）连通子图数的显式闭式。