A spectral excess theorem for nonregular graphs

Let G = (V G,EG) be a connected graph on n vertices, with diameter D, adjacency matrix A, and distance function ∂. Assume that A has d + 1 distinct eigenvalues λ0 > λ1 > · · · > λd with corresponding multiplicities m0 = 1, m1, . . ., md. From the spectrum of G we then define an inner product 〈·, ·〉4 on the vector space Rd[x] of real polynomials of degree at most d. It is well-known that Rd[x] has a unique orthogonal basis p0(x), p1(x), . . . , pd(x), satisfying deg pi(x) = i and 〈pi(x), pi(x)〉4 = pi(λ0) for 0 ≤ i ≤ d. The number kd := |{(u, v)| u, v ∈ V G, ∂(u, v) = d}|/n is called the average excess of G, and the number pd(λ0) is called the spectral excess of G. The spectral excess theorem, proposed by Fiol and Garriga [2], states that kd ≤ pd(λ0) (1)