求流图的1-因子和1-因子连通

文献1]曾用有向图表示一组线性方程,然后利用这个图(称为流图或Coates图)写出方程的解。此法在各方面,尤其是在电路理论方面得到广泛的应用,不过在计算中需要列举流图中全部1-因子(1-factor)和1-因子连通(1-factorial connection),对于较复杂

TO FIND ALL THE 1-FACTORS AND 1-FACTORIAL CONNECTIONS OF A FLOW GRAPH

This note gives a method to find all the 1-factors and 1-factorial connections of a flow graph. Let (D) be the set of all subgraphs of a given diagraph G(V, E) and (H) be the set of all subsets of (D). For h₁, h₂ ∈(H), a multiplication operation being called "star" is denoted by the symbol * and is defined in the following: h₁,*h₂ ={x∪y/x∈h₁, y∈h₂, and deg⁺_x∪y(i)<2, deg^-_x∪y(i)<2}. Theorem Let G(V, E) be a diagraph with vertex set  V={1, 2, …,μ}, and let S_k={(k, t)/(k, t)∈ E, t ∈V}. Then all the 1-factors of G(V,E) can be determined by the product of S_k as follows:C=S₁*S₂*…*S_μ Obviously, if G(V,E) is replaced by G(V, E) is replaced by G(V, E)U(j, i), (j, i) ?E, then the product gives all the 1-factorial connections from em to j of the diagraph G(V, E).