高阶匹配多项式与d-正交性
Higher order matching polynomials and d-orthogonality
摘要：
我们组合地证明了几类图的高阶匹配多项式是d-正交多项式。图的匹配多项式是图被不相交边覆盖的生成函数;高阶匹配多项式对应于路径覆盖。几类经典的正交多项式--切比雪夫多项式、厄米多项式和拉盖尔多项式--可以被解释为路、圈、完全图和完全二部图的匹配多项式。d-正交性的概念是多项式正交性的一般概念的推广，我们使用符号反转对合来证明高阶切比雪夫（第一和第二类），埃尔米特和拉盖尔多项式是d-正交的。我们还调查的时刻，并找到这些多项式的生成函数。
英文摘要：
We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
