Vertex isoperimetry on signed graphs and spectra of non-bipartite Cayley and Cayley sum graphs

For a non-bipartite finite Cayley graph, we show the non-trivial eigenvalues of its normalized adjacency matrix lie in the interval $$[-1+\frac{c{h}_{out}^2}{d},1-\frac{C{h}_{out}^2}{d}],$$ for some absolute constant $c$ and $C$, where $h_{out}$ stands for the outer vertex boundary isoperimetric constant. This improves upon recent obtained estimates aiming at a quantitative version of a result due to Breuillard, Green, Guralnick and Tao. We achieve this by extending the work of Bobkov, Houdré and Tetali on vertex isoperimetry to the setting of signed graphs. We further extend our interval estimate to the settings of vertex transitive graphs and Cayley sum graphs. As a byproduct, we answer positively open questions proposed recently by Moorman, Ralli and Tetali.