Graphs with the same determinant as a complete graph
It is well known that the determinant of the adjacency matrix of a complete graph \(K_n\) is equal to \((-1)^{n-1}(n-1)\). Recently it has been conjectured by \textit{X. Yong} [Linear Algebra Appl. 295, No. 1-3, 73-80 (1999; Zbl 0931.05056)] that \(K_n\) is the only class of graphs with such property. In the paper under review a family of symmetric circulant \((0,1)\)-matrices is studied. Each matrix in the family generates a graph that forms a counterexample to the conjecture. Eigenvalues of these matrices have an interesting octet structure.
scientific article; zbMATH DE number 1485566