For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that r≥min{λ+λ2+4k2,a3} if D = 3 and r≥λ+λ2+4k2 if D ≥ 4, where λ = a1. This result can be generalized to the class of edge-regular graphs. For an edge-regular graph with parameters (v, k, λ) and diameter D ≥ 4, we compare λ+λ2+4k2 with the local valency λ to find a relationship between the second largest eigen-value and the local valency. For an edge-regular graph with diameter 3, we look at the number λ−μ¯+(λ−μ¯)2+4(k−μ¯)2 where μ¯=k(k−1−λ)v−k−1, and compare this number with the local valency λ to give a relationship between the second largest eigenvalue and the local valency. Also, we apply these relationships to distance-regular graphs.

Keywords: edge-regular graphs, distance-regular graphs, second largest eigen-values, local valency.