The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace

Abstract

Two connected labelled graphs H1 and H2 of nullity one, with identical one-vertex deleted subgraphs H1 − z1 and H2 − z2 and having a common eigenvector in the nullspace of their 0-1 adjacency matrix, can be overlaid to produce the superimposition Z. The graph Z is H1 + z2 and also H2 + z1 whereas Z + e is obtained from Z by adding the edge {z1, z2}. We show that the nullity of Z cannot take all the values allowed by interlacing. We propose to classify graphs with two chosen vertices according to the type of the vertices occurring by using a 3-type-code. Out of the 27 values it can take, only 9 are hypothetically possible for Z, 8 of which are known to exist. Moreover, the SSP molecular model predicts conduction or insulation at the Fermi level of energy for 11 possible types of devices consisting of a molecule and two prescribed connecting atoms over a small bias voltage. All 11 molecular device types are realizable for general molecules, but the structure of Z and of Z + e restricts the number to just 5.