On the spectrum and expansion of graph associahedra

Abstract

In this article, we contribute to the spectral analysis of graph associahedra by providing a lower bound for the second largest eigenvalue of $\mathcal{A}(G)$. Furthermore, using equitable partitions, we analyze the spectrum of the stellohedron $\mathcal{A}(K_{1,n})$. Specifically, we prove the existence of an eigenvalue in each interval $(n-i,n-i+1]$ for $1\le i\le 5$, establish the presence of an eigenvalue with high multiplicity in $(n-\frac{3}{n}+\frac{2}{n^2-n},n)$, and identify two additional small eigenvalues.