任意直径的极小族
Diminimal families of arbitrary diameter
摘要：
给定一棵树$T$，设$q(T)$是其基础图为$T$的对称矩阵中不同特征值的最小数目。众所周知，$q(T)\geq d(T)+1$，其中$d(T)$是$T$的直径，如果$q(T)=d(T)+1$，则称树$T$是二极小的。在本文中，我们提出了任何固定直径的双极小树的家庭。我们的证明是建设性的，使我们能够计算，对于任何diminimal树$T$的直径$d$在这些家庭，一个对称矩阵$M$与底层图形$T$的频谱有正是$d+1$不同的特征值。
英文摘要：
Given a tree $T$, let $q(T)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $T$. It is well known that $q(T)\geq d(T)+1$, where $d(T)$ is the diameter of $T$, and a tree $T$ is said to be diminimal if $q(T)=d(T)+1$. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree $T$ of diameter $d$ in these families, a symmetric matrix $M$ with underlying graph $T$ whose spectrum has exactly $d+1$ distinct eigenvalues.
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。
