K_{n,n+1} is the unique graph on 2n+1 vertices with at least n^2 + n edges avoiding equal-degree vertices at path distance 5, for all n >= 11.
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Paths of length five with equal-degree endpoints
K_{n,n+1} is the unique graph on 2n+1 vertices with at least n^2 + n edges avoiding equal-degree vertices at path distance 5, for all n >= 11.