Relation between the H-rank of a mixed graph and the girth of its underlying graph

Let Σ^π=(V(Σ^π),E(Σ^π)) be a mixed graph obtained from a simple graph Γ with the same vertex set V(Γ) and an edge set E(Γ) containing undirected edges and arcs. Let H_A(Σ^π) be the (first kind of) Hermitian adjacency matrix of Σ^π. The H-rank of Σ^π is the rank of H_A(Σ^π), denoted by r^H(Σ^π). The girth of Γ is the length of the shortest cycle in Γ, dented by g(Γ) (or simply by g). In this paper, we show that under some conditions the H-rank of a mixed graph is equal to the girth of its underlying graph. Moreover, we characterize mixed graphs with H-rank g−1 and g+2, distinct from the characterization of T-gain graphs provided by Khan (2024).