On the second eigenvalue of a graph
Discrete Mathematics
Existence and explicit constructions of q+1 regular Ramanujan graphs for every prime power q
Journal of Combinatorial Theory Series B
The size of bipartite graphs with a given girth
Journal of Combinatorial Theory Series B
Hi-index | 0.00 |
For a finite connected graph G let ρ(G˜) be the spectral radius of its universal cover. We prove that ρ(G˜)≥2√d - 1 for any graph G of average degree d ≥ 2 and derive from it the following generalization of the Alon Boppana bound. If the average degree of the graph G after deleting any radius r ≥ 2 ball is at least d ≥ 2, then its second largest eigenvalue in absolute value λ(G) is at least 2 √d - 1(1 - c log r/r). for some absolute constant c. This result is tight in the sense that we can construct graphs with high average degree and diameter but small λ(G).For bipartite graphs with minimal degree at least two, we prove that ρ(G˜)≥√dL - 1 + √dR - 1, where dL, dR are the average degrees on the left and right hand sides.