标题： 一个正则整图族及其在$n$-皇后图中的应用 显示英文标题

标题： A family of regular integral graphs and its application to the $n$-Queens' graph

摘要： 在[ I.F.S. Costa, The$n$-Queens graph and its generalizations, Ph.D. Thesis, University of Aveiro 2024 ]中引入的一类正则积分图，记为${\cal T}(n)$，并在此称为三角形图，进行了分析。在这一分析中，强调了图谱的一致结构和相应特征向量的模式。这些图的性质被研究，并应用于将$n$-Queens' 图分解为三个不同的族：一个由两个三角形图${\cal T}(n)$和${\cal T}(n-1)$组成的单图族，一个由团组成的单图族，以及一个完全二部图族。最后，利用Weyl不等式，我们介绍了一些技术来建立$n$-Queens' 图特征值的下界和上界。 显示英文摘要

摘要： A family of regular integral graphs introduced in [I.F.S. Costa, The $n$-Queens graph and its generalizations, Ph.D. Thesis, University of Aveiro 2024], denoted by ${\cal T}(n)$ and herein called triangular graphs, is analysed. In this analysis, the consistent structure of the graph spectra and the patterns of the corresponding eigenvectors are highlighted. The properties of these graphs are examined and applied to the decomposition of the $n$-Queens' graph into three distinct families: a family of a single graph whose components are two triangular graphs, ${\cal T}(n)$ and ${\cal T}(n-1)$, a family of a single graph whose components are cliques and a family of complete bipartite graphs. Finally, using Weyl's inequalities, we introduce some techniques to establish lower and upper bounds on the eigenvalues of the $n$-Queens' graph.