A lower bound on the spectral radius of the universal cover of a graph

For a finite connected graph G let ρ(G̃) be the spectral radius of its universal cover. We prove that ρ(G̃)⩾2d−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 2d−1(1−c log rr) 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). partite graphs with minimal degree at least two, we prove that ρ(G̃)⩾dL−1+dR−1, where dL, dR are the average degrees on the left and right hand sides.