Some Results on the Eigenvalues of Distance-Regular Graphs

In 1986, Terwilliger showed that there is a strong relation between the eigenvalues of a distance-regular graph and the eigenvalues of a local graph. In particular, he showed that the eigenvalues of a local graph are bounded in terms of the eigenvalues of a distance-regular graph, and he also showed that if an eigenvalue θ of the distance-regular graph has multiplicity m less than its valency k, then -1-b1θ+1 is an eigenvalue for any local graph with multiplicity at least k-m. In this paper, we are going to generalize the results of Terwilliger to a broader class of subgraphs. Instead of local graphs, we consider induced subgraphs of distance-regular graphs (and more generally t-walk-regular graphs with t a positive integer) which satisfies certain regularity conditions. Then we apply the results to obtain bounds on eigenvalues of t-walk-regular graphs if the girth equals 3, 4, 5, 6 or 8. In particular, we will show that the second largest eigenvalue of a distance-regular graph with girth 6 and valency k is at most k-1, and we will show that the only such graphs having k-1 as its second largest eigenvalue are the doubled Odd graphs.