An algorithm to evaluate the spectral expansion
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Assume that X is a connected (q+1)-regular undirected graph of finite order n. Let A denote the adjacency matrix of X. Let λ_1=q+1>λ_2≥λ_3≥...≥λ_n denote the eigenvalues of A. The spectral expansion of X is defined by Δ(X)=q+1-max_2≤ i≤ n|λ_i|. By Alon–Boppana theorem, when n is sufficiently large, Δ(X) is quite high if μ(X)=q^-1/2max_2≤ i≤ n|λ_i| is close to 2. In this paper we introduce a number sequence {H_k}_k=1^∞ and study its connection with μ(X). Furthermore, with the inputs A and a real number ε>0 we design an algorithm to estimate if μ(X)≤ 2+ε in O(n^ωloglog_1+ε n ) time, where ω<2.3729 is the exponent of matrix multiplication.
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