计算有向图线性不变量的算法
Algorithms for computing linear invariants if directed graphs
摘要：
称域$K$，\emph{simultaneously K-equivalent}上的$m\times n$矩阵的两对$(M,N)$和$(M',N')$，如果存在K上的平方可逆矩阵$S,T$，其中有$M'=SMT$和$N'=SNT$ . Kronecker \cite{Kronecker}给出了矩阵对同时等价的一组完整的不变量。以自然的方式关联到有限有向图$Γ$，具有$v$顶点和$e$边，0和1的$e\times v$矩阵的有序对$(M,N)$。很自然地，我们会尝试计算这样一对$(M,N)$的克罗内克不变量，特别是因为它们清楚地提供了$Γ$的同构不变量。当两个对应的图对同时等价时，我们称这两个图为“线性等价的”。本文的目的是从有限有向图的基本组合性质出发，直接计算有限有向图的Kronecker不变量。令人惊喜的是，这些新的不变量是纯粹的{PDMATHAMBMZ有理} -事实上，{PDMATHAMBMZ积分}，在这个意义上，判定两个有向图是否线性等价所需的计算仅涉及对从每个给定图构造的各种有限图中的顶点进行计数--并且{PDMATHAMBMZ不涉及找到K上多项式的不可约因子分解}（在明显的对比都熟悉的不变量计算图提供的特征值的连接矩阵，和同构问题的一般对矩阵。
英文摘要：
Call two pairs $(M,N)$ and $(M',N')$ of $m\times n$ matrices over a field $K$, \emph{simultaneously K-equivalent} if there exist square invertible matrices $S,T$ over K, with $M'=SMT$ and $N'=SNT$. Kronecker \cite{Kronecker} has given a complete set of invariants for simultaneous equivalence of pairs of matrices. Associate in the natural way to a finite directed graph $Γ$, with $v$ vertices and $e$ edges, an ordered pair $(M,N)$ of $e\times v$ matrices of zeros and ones. It is natural to try to compute the Kronecker invariants of such a pair $(M,N)$, particularly since they clearly furnish isomorphism-invariants of $Γ$. Let us call two graphs `linearly equivalent' when their two corresponding pairs are simultaneously equivalent. The purpose of the present paper, is to compute directly these Kronecker invariants of finite directed graphs, from elementary combinatorial properties of the graphs. A pleasant surprise is that these new invariants are purely {\bf rational} --indeed, {\bf integral}, in the sense that the computation needed to decide if two directed graphs are linearly equivalent only involves counting vertices in various finite graphs constructed from each of the given graphs-- and {\bf does not involve finding the irreducible factorization of a polynomial over K} (in apparent contrast both to the familiar invariant-computations of graphs furnished by the eigenvalues of the connection matrix, and to the isomorphism problem for general pairs of matrices.)
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