图的矩阵表示及其谱作为图不变量
A matrix representation of graphs and its spectrum as a graph invariant
摘要：
我们使用线有向图的构造来将正交矩阵与每个图相关联。从这个正交矩阵，我们得到另外两个矩阵。这三个矩阵中的每一个的谱被认为是图不变量。对于前两种情况，我们明确地计算了谱，并表明它是由原始图的邻接矩阵的谱决定的。然后，我们通过计算表明，同构类的许多已知的家庭的强正则图（高达64个顶点）的特点是这个矩阵的谱。我们猜想，这总是强正则图的情况下，我们证明了猜想是不成立的一般图。我们验证了我们的方法无法区分的最小正则图是在14个顶点上。
英文摘要：
We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show by computation that the isomorphism classes of many known families of strongly regular graphs (up to 64 vertices) are characterized by the spectrum of this matrix. We conjecture that this is always the case for strongly regular graphs and we show that the conjecture is not valid for general graphs. We verify that the smallest regular graphs which are not distinguished with our method are on 14 vertices.
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