2025/04/25 by Wei Wang, Lin, Limeng, Hao Zhang +2
Computer Science · Mathematics · Physics and Astronomy · #05C50 #Combinatorics (math.CO) #Complex Network Analysis Techniques #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2504.18079
openalex publication_date 2025/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper examines the spectral characterizations of oriented graphs. Let Σ be an n-vertex oriented graph with skew-adjacency matrix S. Previous research mainly focused on self-converse oriented graphs, proposing arithmetic conditions for these graphs to be uniquely determined by their generalized skew-spectrum (DGSS). However, self-converse graphs are extremely rare; this paper considers a more general class of oriented graphs Gn (not limited to self-converse graphs), consisting of all n-vertex oriented graphs Σ such that 2- \lfloor (n)/(2) \rfloor det W(Σ) is an odd and square-free integer, where W(Σ)=[e,Se,…,Sn-1e] (e is the all-one vector) is the skew-walk matrix of Σ. Given that Σ is cospectral with its converse Σ\rm T, there always exists a unique regular rational orthogonal Q0 such that Q0\rm TSQ0=-S. This study reveals that there exists a deep relationship between the level ℓ0 of Q0 and the number of generalized cospectral mates of Σ. More precisely, we show, among others, that the maximum number of generalized cospectral mates of Σ\inGn is at most 2t-1, where t is the number of prime factors of ℓ0. Moreover, some numerical examples are also provided to demonstrate that the above upper bound is attainable. Finally, we also provide a criterion for the oriented graphs Σ\inGn to be weakly determined by the generalized skew-spectrum (WDGSS).