Abstract
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 .
Highlights
The graphs we consider are simple, that is they are undirected with no multiple edges or loops
The 0-1 adjacency matrix G = of a labelled graph G on n vertices is a n × n matrix such that aij = 1 if there is an edge between the vertices i and j, and aij = 0 otherwise
Following the terminology used in [4], a vertex u is a core vertex (CV), a middle core-forbidden vertex (CFVmid) or an upper coreforbidden vertex (CFVupp) if the nullity of the graph G − u obtained from G upon deleting the vertex u is η(G) − 1, η(G), or η(G) + 1, respectively
Summary
The graphs we consider are simple, that is they are undirected with no multiple edges or loops. Following the terminology used in [4], a vertex u is a CV, a middle core-forbidden vertex (CFVmid) or an upper coreforbidden vertex (CFVupp) if the nullity of the graph G − u obtained from G upon deleting the vertex u is η(G) − 1, η(G), or η(G) + 1, respectively. It follows from Proposition 1.1 that CFVs are vertices corresponding to a zero entry in each kernel eigenvector in the nullspace of G.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
