三次Lehman矩阵的结构
Structure of Cubic Lehman Matrices
摘要：
如果$AB^T=J+kI$对某个整数$k\in\{-1,1,2,3,\ldots\}$是$(0,1)$ -矩阵的一对$(A,B)$，则称之为\emph{Lehman pair}.在这种情况下，$A$和$B$称为\emph{Lehman matrices}。这个术语的出现是因为雷曼证明了在任何非退化的极小非理想（mni）矩阵$M$中具有最少行的行形成$M$的方形雷曼子矩阵。具有$k=-1$的莱曼矩阵本质上等价于\emph{partitionable graphs}（也称为$(α,ω)$ -图），因此已经被大量研究作为直接分类最小不完美图的尝试的一部分。在本文中，我们认为莱曼矩阵的二部邻接矩阵的一个规则的二部图，特别关注的情况下，该图是立方。从这个角度来看，我们确定了两个结构，从较小的雷曼图生成立方雷曼图。这些构造中最多产的是用一个特定的$3$-顶点子图（我们称之为$3$ -梯级梯段）反复替换合适的边对。二十年前，Lütolf \& Margot开始了mni矩阵的计算研究，并构建了一个目录，其中包含（除其他事项外）所有三次雷曼矩阵的列表，$k =1$的顺序高达$17 \times 17$。我们验证了他们的目录（只有一个遗漏），并将计算结果推广到$20 \times 20$矩阵。$908$立方雷曼矩阵（与$k=1$）的顺序高达$20 \times 20$，只有两个不产生于我们的$3$ -横档阶梯建设。然而，这些例外情况可以从我们的第二个结构中导出，因此我们的两个结构覆盖了所有已知的具有$k=1$的三次雷曼矩阵。
英文摘要：
A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called \emph{Lehman matrices}. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs} (also known as $(α,ω)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, Lütolf \& Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$.
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