2021/03/25 by Monu Kadyan, Bikash Bhattacharjya, Kadyan, Monu +1
Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2103.13632
openalex publication_date 2021/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let 0 ∈ Γ and Γ∖ \0\ be a subgroup of the complex numbers of unit modulus. Define Hn(Γ) to be the set of all n× n Hermitian matrices with entries in Γ, whose diagonal entries are zero. The matrices A,B∈ Hn(Γ) are said to be switching equivalent if there is a diagonal matrix D, in which the diagonal entries belong to Γ∖ \0\, such that D-1 A D=B. We find a characterization, in terms of fundamental cycles of graphs, of switching equivalence of matrices in Hn(Γ). We give sufficient conditions to characterize the cospectral matrices in Hn(Γ). We find bounds on the number of switching equivalence classes of all mixed graphs with the same underlying graph. We also provide the size of all switching equivalence classes of mixed cycles, and give a formula that calculates the size of a switching equivalence class of a mixed plane graph. We also discuss an action of the automorphism group of a graph on switching equivalence classes of matrices in Hn(Γ).