Let λ 1 be the largest eigenvalue of the matrix A (which is the adjacency matrix of a graph G).We note in passing that since G is connected, λ 1 is simple (by using Perron-Frobenius theorem). Then A/λ1 has spectral radius 1.
Even without knowing that λ 1 is simple, it is clear that “A/λ1” has spectral radius 1. Am I wrong?