TY - JOUR
T1 - Some mixed graphs with H-rank 4, 6 or 8
AU - Yang, Jinling
AU - Wang, Ligong
AU - Yang, Xiuwen
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.
PY - 2021/4
Y1 - 2021/4
N2 - The H-rank of a mixed graph Gα is defined to be the rank of its Hermitian adjacency matrix H(Gα). If Gα is switching equivalent to a mixed graph (Gα)′, and two vertices u, v of Gα have exactly the same neighborhood in (Gα)′, then u and v are said to be twins. The twin reduction graph TGα of Gα is a mixed graph whose vertices are the equivalence classes, and [u][v]∈E(TGα) if uv∈E((Gα)′), 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).
AB - The H-rank of a mixed graph Gα is defined to be the rank of its Hermitian adjacency matrix H(Gα). If Gα is switching equivalent to a mixed graph (Gα)′, and two vertices u, v of Gα have exactly the same neighborhood in (Gα)′, then u and v are said to be twins. The twin reduction graph TGα of Gα is a mixed graph whose vertices are the equivalence classes, and [u][v]∈E(TGα) if uv∈E((Gα)′), 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).
KW - H-rank
KW - Mixed bipartite graph
KW - Switching equivalence
UR - http://www.scopus.com/inward/record.url?scp=85100789557&partnerID=8YFLogxK
U2 - 10.1007/s10878-021-00704-6
DO - 10.1007/s10878-021-00704-6
M3 - 文章
AN - SCOPUS:85100789557
SN - 1382-6905
VL - 41
SP - 678
EP - 693
JO - Journal of Combinatorial Optimization
JF - Journal of Combinatorial Optimization
IS - 3
ER -