Counting the geodesic cycles of a given length
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Assume that X is a connected regular undirected graph of finite order n. Let N_k denote the number of geodesic cycles on X of length k. The numbers {N_k}_k=1^∞ first appeared in the Ihara zeta function of X. The Hasse–Weil bounds on {N_k}_k=1^∞ provide a necessary and sufficient condition for X as a Ramanujan graph. For a given k, we propose a fast algorithm that computes the number N_k in O(n^ω k ) time, where ω<2.3727 is the exponent of matrix multiplication.
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