Some mixed graphs with H-rank 4, 6 or 8

The H-rank of a mixed graph $$G^{\alpha }$$ G α is defined to be the rank of its Hermitian adjacency matrix $$H(G^{\alpha })$$ H ( G α ) . If $$ G^{\alpha } $$ G α is switching equivalent to a mixed graph $$(G^{\alpha })' $$ ( G α ) ′ , and two vertices u, v of $$G^{\alpha }$$ G α have exactly the same neighborhood in $$(G^{\alpha })'$$ ( G α ) ′ , then u and v are said to be twins. The twin reduction graph $$T_{G^{\alpha }}$$ T G α of $$G^{\alpha }$$ G α is a mixed graph whose vertices are the equivalence classes, and $$[u][v]\in E(T_{G^{\alpha }})$$ [ u ] [ v ] ∈ E ( T G α ) if $$uv\in E((G^{\alpha })')$$ u v ∈ E ( ( G α ) ′ ) , where [u] denotes the equivalence class containing the vertex u. In this paper, we give the upper (resp., lower) bound of the number of vertices of the twin reduction graphs of connected mixed bipartite graphs, and characterize all twin reduction graphs of the connected mixed bipartite graphs with H-rank 4 (resp., 6 or 8). Then, we characterize all connected mixed graphs with H-rank 4 (resp., 6 or 8) among all mixed graphs containing induced mixed odd cycles whose lengths are no less than 5 (resp., 7 or 9).