The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace - Sciriha, Irene (Ars mathematica contemporanea, 2019)
Two connected labelled graphs ?$H_1$? and ?$H_2$? of nullity one, with identical one-vertex deleted subgraphs ?$H_1 - z_1$? and ?$H_2 - z_2$? 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 ?$H_1 + z_2$? and also ?$H_2 + z_1$? whereas ?$Z + e$? is obtained from ?$Z$? by adding the edge ?$\{z_1, z_2\}$?. 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.
Keywords: circuit; core vertices; key-graphs; ključni grafi; ničnost; nullity; prekrivanje; superimposition; vezje; vozlišča sredice