The H-rank of a mixed graph \(G^{\alpha }\) is defined to be the rank of its Hermitian adjacency matrix \(H(G^{\alpha })\). If \( G^{\alpha } \) is switching equivalent to a mixed graph \((G^{\alpha })' \), and two vertices u, v of \(G^{\alpha }\) have exactly the same neighborhood in \((G^{\alpha })'\), then u and v are said to be twins. The twin reduction graph \(T_{G^{\alpha }}\) of \(G^{\alpha }\) is a mixed graph whose vertices are the equivalence classes, and \([u][v]\in E(T_{G^{\alpha }})\) if \(uv\in E((G^{\alpha })')\), 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).

所述ħ秩的混合图的\（G ^ {\阿尔法} \）被定义为它的厄密邻接矩阵的秩\（H（G ^ {\阿尔法}）\） 。如果\（G ^ {\阿尔法} \）被切换相当于混合图\（（G ^ {\阿尔法}）” \），和两个顶点ü，  v的\（G ^ {\阿尔法} \）具有\（（G ^ {\ alpha}）'\）中完全相同的邻域，则称u和v为双胞胎。双子减少图表\（T_ {G ^ {\阿尔法}} \）的\（G ^ {\阿尔法} \）是一个混合图，其顶点是等价类，和\（[u] [v] \ in E（T_ {G ^ {\ alpha}}）\） if \（uv \ in E（（G ^ {\ alpha}}'）\），其中[ u ]表示包含顶点u的等价类。在本文中，我们给出了连通混合二分图的孪生约化图的顶点数目的上限（分别为下限），并用H -rank 4（resp）表征了连通混合二分图的所有孪生约化图的特征。。，6或8）。然后，我们在所有包含诱导混合奇数周期且长度不小于5（分别为7或9）的混合图中，用H- 4级（分别为6或8）来表征所有连通混合图。