Common Eigenvalues of Vertex-Decorated Regular Graphs

Let $G=(V,E)$ be a simple graph with the vertex set V and the edge set E$\left(\left|V\right|=n,\left|E\right|=m\right)$. An example of a vertex-decorated graph $DG$ is a vertex-quadrangulated graph $QG$. The vertex quadrangulation $QG$ of 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. If we contract each quadrangle of $QG$ to a point that takes over the incidence of the four edges that were previously joined to this quadrangle, then we can again get the original graph G. Any connected graph H that provides (some of) its vertices for external connections can play the role of a decorating graph, and any graph G with vertices of valency no greater than the number of contact vertices in H can be decorated with it. Herein, we consider the case when G is a regular graph. Since the decoration also depends on the way the edges are attached to the decorating graph, we clearly stipulate it. We show that all similarly decorated regular graphs $DG$ that meet our conditions have at least $\left|V\right(H\left)\right|$ predicted common eigenvalues. A number of related results are proven. As possible applications of these results in chemistry, cases of simplified findings of eigenvalues of a molecular graph even in the absence of the usual symmetry of the molecule may be of interest. This, in particular, can somewhat expand the possibilities of applying the simple Hückel method for large molecules.