$k$-spectrally monomorphic tournaments

TL;DR

This paper characterizes $k$-spectrally monomorphic tournaments, showing that non-transitive ones are highly restricted and linking certain spectral properties to doubly regular tournaments.

Contribution

It provides a complete classification of $k$-spectrally monomorphic tournaments for various $k$, revealing their structure and relation to doubly regular tournaments.

Findings

Transitive tournaments are trivially $k$-spectrally monomorphic.

No non-transitive $k$-spectrally monomorphic tournaments exist for $k otin igrace{ ext{3, ..., } n-3}$.

Non-transitive $(n-2)$-spectrally monomorphic tournaments are exactly the doubly regular tournaments.

Abstract

A tournament is $k$-spectrally monomorphic if all the $k\times k$ principal submatrices of its adjacency matrix have the same characteristic polynomial. Transitive $n$-tournaments are trivially $k$-spectrally monomorphic. We show that there are no other for $k\in \{3,\ldots,n-3\} $. Furthermore, we prove that for $n\geq 5$, a non-transitive $n$-tournament is $(n-2)$-spectrally monomorphic if and only if it is doubly regular. Finally, we give some results on $(n-1)$-spectrally monomorphic regular tournaments.