Switching equivalence of Hermitian adjacency matrix of mixed graphs

Monu Kadyan, Bikash Bhattacharjya

Published 2021-03-25Version 1

Let $0 \in \Gamma$ and $\Gamma \setminus \{0\}$ be an abelian group under multiplication, where $\Gamma \setminus \{0\} \subseteq \{ z\in \mathbb{C}: |z|=1 \}$. Define $\mathcal{H}_{n}(\Gamma)$ to be the set of all $n\times n$ Hermitian matrices with entries in $\Gamma$, whose diagonal entries are zero. We introduce the notion of switching equivalence on $\mathcal{H}_{n}(\Gamma)$. We find a characterization, in terms of fundamental cycles of graphs, of switching equivalence of matrices in $\mathcal{H}_{n}(\Gamma)$. We give sufficient conditions to characterize the cospectral matrices in $\mathcal{H}_{n}(\Gamma)$. 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 planar graph. We also discuss the action of automorphism group of a graph on switching equivalence classes.