Potential counter-examples to a conjecture on the column space of the adjacency matrix
Year: 2025
Type: article
Abstract: Attempts to resolve the Akbari-Cameron-Khosrovshahi-conjecture have so far focused on the rank of a matrix. The conjecture claims that there exists a nonzero (0, 1)-vector in the row space of a (0, 1)-adjacency matrix A {\bf{A}} of a graph G G , that is not a copy of any row of A {\bf{A}} . We present a new approach different from the methods used to date. By considering the change in the nullity of A {\bf{A}} on adding an arbitrary vertex to a base graph G G , we seek counter-examples to the conjecture. As a result, we determine a class C {\mathcal{C}} of graphs that could be potential counter-examples to the conjecture. We use eigenvector techniques to show that C {\mathcal{C}} is restricted to the intersection of a number of families of graphs with particular properties.
Source: Special Matrices
Institution: University of Malta
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,187
OpenAlex ID: W4409328076