AI 中文总结
本文结合插值性质与 graft 的局部补小子式，刻画了具有二项式部分 Petrial 多项式的连通简单图必为路径，同时明确树具有三项式该多项式当且仅当为 T 形或 H 形树，并推导了相关显式公式。
AI 中文摘要
花束的部分 Petrial 多项式可通过其相交图邻接矩阵对角元变化得到的矩阵在 GF(2) 上的余秩计算得出。受此矩阵形式的启发，我们研究简单图对应的多项式，并确定其恰有两个或三个非零项的情况。一个关键工具是插值性质：非零项的指数是连续的。结合该性质与 graft 的局部补小子式，我们将二项式情形的已知刻画从连通圆图扩展至所有连通简单图，证明此类图具有二项式部分 Petrial 多项式当且仅当它是一条路径。我们的主要结果刻画了树的三项式情形：树具有三项式部分 Petrial 多项式当且仅当它是 T 形树或 H 形树。其中，T 形树的最大度为 3，且恰有一个度为 3 的顶点；H 形树的最大度为 3，且恰有两个相邻的度为 3 的顶点。我们还根据 Jacobsthal 数推导出了这两类树的显式公式。
英文摘要
The partial Petrial polynomial of a bouquet can be computed from the coranks over GF(2) of matrices obtained by varying the diagonal entries of the adjacency matrix of its intersection graph. Motivated by this matrix formulation, we study the corresponding polynomial for simple graphs and determine when it has exactly two or three nonzero terms. A key tool is the interpolating property: the exponents of the nonzero terms are consecutive. Using this property together with local complementation minors of grafts, we extend the known characterization of the binomial case from connected circle graphs to all connected simple graphs, showing that such a graph has a binomial partial Petrial polynomial if and only if it is a path. Our main result characterizes the trinomial case for trees: a tree has a trinomial partial Petrial polynomial if and only if it is a T-shape tree or an H-shape tree. Here, a T-shape tree has maximum degree 3 and exactly one vertex of degree 3, whereas an H-shape tree has maximum degree 3 and exactly two vertices of degree 3, which are adjacent. We also derive explicit formulas for both families in terms of Jacobsthal numbers.
Comments15 pages,2 figures
