Eigenvalue −1 of the Vertex Quadrangulation of a 4-Regular Graph

The vertex quadrangulation $QG$ of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the question was posed: does the spectrum of an arbitrary unweighted graph $QG$ include the full spectrum $\{3,{(-1)}^3\}$ of the tetrahedron graph (complete graph $K_4$)? Previously, many bipartite and nonbipartite graphs $QG$ with such a subspectrum have been found; for example, a nonbipartite variant of the graph $Q{K}_5$. Here, we present one of the variants of the nonbipartite vertex quadrangulation $QO$ of the octahedron graph O, which has eigenvalue $(-1)$ of multiplicity 2 in the spectrum, while the spectrum of the bipartite variant $QO$ contains eigenvalue $(-1)$ of multiplicity 3. Thus, in the case of nonbipartite graphs, the answer to the question posed depends on the particular graph $QG$. Here, we continue to explore the spectrum of graphs $QG$. Some possible connections of the mathematical theme to chemistry are also noted.