In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply our results to obtain a lower bound on the smallest eigenvalue of the associahedron graph, and we prove that this bound gives the correct order of magnitude of this eigenvalue. We also survey what is known regarding the second-largest eigenvalue of the associahedron graph.
翻译:在本文中,我们获得了一个包含许多小固定子集的普通图中最小的精度值的下限。 这概括了Aharoni、 Alon 和 Berger 的结果, 该子集是一个三角形。 我们应用我们的结果来获得一个最小的精度值的精度值。 我们证明这个精度值的精度顺序是正确的。 我们还调查了已知的亚精度图中第二大精度值的精度值。