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 product G(H) of graphs G and H. Let $$F = (V, E)$$ be a molecular graph with the vertex set V and the edge set E $$(|V|=n; |E|=m)$$ , 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.