The First Eigenvalue of (c,d)-Regular Graph

Abstract

We show a phase transition of the first eigenvalue of random (c,d)-regular graphs, whose instance of them consists of one vertex with degree c and the other vertices with degree d for c>d. We investigate a reduction from the first eigenvalue analysis of a general (c,d)-regular graph to that of a tree, and prove that, for any fixed c and d, and for a graph G chosen from the set of all (c,d)-regular graphs with n vertices uniformly at random, the first eigenvalue of G is approximately $\max \{d,\, c / \sqrt{c-d+1} \}$ with high probability.