A graph is called a chain graph if it is bipartite and the neighbourhoods of the vertices in each colour class form a chain with respect to inclusion. In this paper we give an explicit formula for the characteristic polynomial of any chain graph and we show that it can be expressed using the determinant of a particular tridiagonal matrix. Then this fact is applied to show that in a certain interval a chain graph does not have any nonzero eigenvalue. A similar result is provided for threshold graphs.

中文翻译：

---

一些图类的三对角矩阵和谱属性

如果一个图是二部图，并且每个颜色类中顶点的邻域形成一个关于包含的链，则该图称为链图。在本文中，我们给出了任何链图的特征多项式的显式公式，并表明它可以使用特定三对角矩阵的行列式来表示。然后应用这个事实来证明在某个区间内，链图没有任何非零特征值。为阈值图提供了类似的结果。