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

Abstract

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.