秩为4的二部符号混合图

Bipartite Signed Mixed Graphs with Rank 4

符号混合图是一个混合图通过给每条边指定一个符号所得的图,是符号图和混合图一个自然的推广.符号混合图的秩是指其对应的Eisenstein邻接矩阵的秩.Wissing和van Dam在文[J.Combin.Theory Ser.A,2022,187:105573,40 pp.]中确定了所有秩为2和3的符号混合图.本文刻画了秩为4的二部符号混合图,部分解决了该文所提的第三个公开问题.

A signed mixed graph is obtained by assigning a sign to each edge of a mixed graph,which is a natural generalization of signed graphs and mixed graphs.The rank of signed mixed graph refers to the rank of its Eisenstein adjacency matrix.In[J.Combin.Theory Ser.A,2022,187:105573,40 pp.],Wissing and van Dam characterized all signed mixed graphs with rank 2 and 3.In this paper,we characterize all bipartite signed mixed graphs with rank 4,which partially answers the third open problem in the paper above.