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.