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

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]is an element of E(T-G alpha) if uv is an element of 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).

Automated summary

This paper explores the phenomenon of twins in mixed graphs and the properties of twin reduction graphs, providing upper and lower bounds for the number of vertices in certain cases and characterizing twin reduction graphs for connected mixed bipartite graphs with specific H-ranks. It also identifies all connected mixed graphs with certain H-ranks among those containing induced mixed odd cycles of certain lengths.