A Relationship between the Second Largest Eigenvalue and Local Valency of an Edge-regular Graph

For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that (Formula Presented) if D = 3 and (Formula Presented) 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 (Formula Presented) 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 (Formula Presented), where (Formula Presented), 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.