恰有一公共点的双圈图的邻接矩阵的奇异性

Singularity of Adjacency Matrix of Graphs with Exactly One Copoint Bicycle

恰有一公共点的双圈图的邻接矩阵是奇异的当且仅当G满足:G有完美匹配,c1与c2中一个是4 m圈,另一个是偶圈,4 m圈上不挂出奇数阶树;G有完美匹配,G-V(c1)-V(c2)含完美匹配,G-V(c1)或G-V(c2)含完美匹配,且含有4 m圈;G无完美匹配,G-V(c1)和G-V(c2)均含有完美匹配,且G中含有4k1+3和4e1+1(k1,e1∈N)阶图;G,G-V(c1)和G-V(c2)都不含完美匹配.恰有一公共点的双圈图的邻接矩阵的行列式的最大值是4.

The adjacency matrix of a graph with exactly one co - point bicyclesis singular if and only if G has a perfect matching and a cycle of order 4 m （ m ∈ N） and the other cycle of order 2k（k∈N）, and there isn＇t an odd tree under the cycle of order 4 m; or G and G - V （c1 ） - V （c2 ） and one of G - V（c1 ） and G - V（c2 ） all have perfect matching and has a cycle of order 4 m（m∈N） ; or if G has no perfect matching, G- V（c1） and G - V（ c2 ） both have a perfect matching, and G has cycles of order 4k1 ＋ 3 and 4e1 ＋ 1 （ k1 , e1 ∈ N） ; or if G and G - V（c1 ） and G - V（ c2 ） all have no perfect matching. The maximum determinant of the adjacency matrix of graphs with only one co - point bicycle is 4.