Construction of \(r\)-regular singular graphs
A graph \(G\) is singular if its adjacency matrix has 0 as an eigenvalue. Let \({\mathcal G}(n,r)\) denote the class of all connected \(r\)-regular graphs on \(n\) vertices. It is known that the only graph \(C_n\) in \({\mathcal G}(n,2)\) is singular if and only if \(n\equiv 0\text{ mod 4}\) and that the only graph \(K_n\) from \({\mathcal G}(n,n-1)\) is not singular. The main result of the paper under review is a construction of a singular graph in \({\mathcal G}(n,r)\) for every \(r>2\) and \(n>r+1\) such that \({\mathcal G}(n,r)\neq\varnothing\). Remarks of the reviewer: The result of the paper seems to be quite trivial. The statement at the end of page~499 that isomorphic graphs could have non-zero determinants of opposite signs is evidently wrong.
scientific article; zbMATH DE number 1484559