No vgr(n,k,5, k(k-1)^2/2 - ε) graphs exist for k ≥ 3 and 0 < ε ≤ (k-1)/2.
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A non-existence result for vertex-girth-regular graphs
No vgr(n,k,5, k(k-1)^2/2 - ε) graphs exist for k ≥ 3 and 0 < ε ≤ (k-1)/2.