Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it is isomorphic to each digraph to which it is cospectral. Convincing numerical evidence to support the claim that this property is extremely rare is provided. Nonetheless, the first infinite family of connected digraphs that is SHDS is constructed. This family is obtained via the introduction of twin vertices into a structure that is named negative tetrahedron. This special digraph, that exhibits extreme spectral behavior, is contained in the surprisingly small collection of all digraphs with exactly one negative eigenvalue, which is determined as an intermediate result.
|Journal||Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics|
|Publication status||Published - Jul 2020|
- Hermitian adjacency matrix
- spectra of digraphs
- cospectral digraphs
- characterization of digraphs
Wissing, P., & van Dam, E. (2020). The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum. Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics, 173, . https://doi.org/10.1016/j.jcta.2020.105232