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A graph is asymmetric if the automorphism group of its set of vertices is trivial. A graph is called non-asymmetric if and only if it is not asymmetric. A graph \( G \) is minimally non-asymmetric if \( G \) is non-asymmetric but \( G – e \) is asymmetric for any edge \( e \) contained in \( G \).
Given a finite set \( V \) (of elements called varieties) and integers \( k \), \( r \), and \( \lambda \), a balanced incomplete block design (BIBD) is a family of \( k \)-element subsets of \( V \), called blocks, such that any element is contained in \( r \) blocks and any pair of distinct varieties \( u \) and \( w \) is contained in exactly \( \lambda \) blocks.
In this paper, we give examples of minimally non-asymmetric graphs constructed from balanced incomplete block designs.