In this paper we study the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we show that in the case of constant branching, the eigenvalues are realized as the roots of a family of generalized Fibonacci polynomials and produce a limiting distribution for the eigenvalues as the tree depth goes to infinity. We indicate how these results can be extended to periodic branching patterns and also provide a generalization to higher order simplicial complexes.

中文翻译：

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关于有限的、有根的同质树的谱

在本文中，我们研究了具有规则分支​​特性的有限根树家族的邻接谱。特别是，我们表明，在恒定分支的情况下，特征值被实现为一组广义斐波纳契多项式的根，并随着树的深度趋于无穷大而产生特征值的极限分布。我们指出如何将这些结果扩展到周期性分支模式，并提供对高阶单纯复形的概括。