Abstract
If the inverse of a non-singular real symmetric matrix that represents an edge-weighted graph with no loops has zero diagonal, then the inverse itself is the matrix of a loopless graph. Here we show that such graphs having non--zero weight on each edge always exist if their number of vertices is at least 6.
Highlights
Since the discovery of fullerenes in 1985, an intimate relationship has matured between these carbon molecules and mathematics
For a graph not to be a NSSD for any associated adjacency matrix, it suffices to show that it has a non–singular vertex–deleted subgraph
We have shown that complete NSSDs exist for all positive integers at least six
Summary
Since the discovery of fullerenes in 1985, an intimate relationship has matured between these carbon molecules and mathematics. Over the years, non–traditional fullerenes composed of cycles of other sizes, which are general 3–regular polyhedra, have been studied, especially for their regularity properties. This current work is motivated mainly by the beauty of these structures, coupled with the elegance of the interplay between combinatorics, graph theory and algebra. This family of graphs has remarkable properties as explored in [2] and [8] It is the purpose of this work to determine the existence of complete NSSDs. In [8], a characterisation of a graph G which is G–nutful is given. Some of these G–nutful graphs turn out to be complete, in which case both G and the graph of G−1 are nutful
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.
