Close-to-zero eigenvalues of the rooted product of graphs

The construction of vertex-decorated graphs can be used to produce derived graphs with specific eigenvalues from undecorated graphs, which themselves do not have such eigenvalues. An instance of a decorated graph is the rooted productG(H) of graphs G and H. Let F = (V, E) be a molecular graph with the vertex set V and the edge set E(vertical bar V vertical bar=n;vertical bar E vertical bar = m), and let n(+) = n(-)(n(+) + n(-)= n), where n(+) and n(-) are the numbers of positive and negative eigenvalues, respectively. Then, in the spectrum of the eigenvalues of F, two minimum-modulus eigenvalues, positive lambda(+) and negative lambda(-), are of special interest because the value delta=lambda(+)-lambda(-) determines the energy gap. In quantum chemistry, the energy gap delta is associated with the energy of an electron transfer from the highest occupied molecular orbital to the lowest unoccupied molecular orbital of a molecule. As an example, we consider obtaining a (molecular) graph F=G(H) whose median eigenvalues lambda(+) and lambda(-) are predictably close to 0.