论文标题
解决Babai问题的解决方案，具有不可用的邻接矩阵
A solution to Babai's problem on digraphs with non-diagonalizable adjacency matrix
论文作者
论文摘要
每个有限图的邻接矩阵都是可对角度化的事实，在光谱图理论中起着基本作用。由于这一事实总体上不适合Digraphs，因此很自然地询问它是否适用于具有一定程度的对称性的挖掘物。对这个问题的兴趣可以追溯到1980年代初，当时P.〜J。〜Cameron要求存在具有不可用的邻接矩阵的弧形传输挖掘物。 L.〜Babai于1985年在肯定的肯定中回答了这一点。然后，巴比提出了构建2个ARC传播的挖掘和一个顶点主要的挖掘物的开放问题，其邻接矩阵不可对角线。在本文中，我们通过为每个整数$ s \ geq2 $和一个无限的顶点主要的挖掘者构建了一个无限的家族，解决了Babai的问题。
The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest on this question dates back to early 1980s, when P.~J.~Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by L.~Babai in 1985. Then Babai posed the open problem of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai's problem by constructing an infinite family of $s$-arc-transitive digraphs for each integer $s\geq2$, and an infinite family of vertex-primitive digraphs, respectively, both of whose adjacency matrices are non-diagonalizable.
