THE McKAY-MILLER-ŠIRÁŇ NGRAPHS AND LIFTS OF AUTOMORPHISMS

Jana Šiagiová

   McKay, Miller, Širáň [2] discovered a family of vertex-transitive graphs G_d of degree d=(3q-1)/2 for prime powers q ≡1(mod 4), diameter 2, and order 8/9(d+1/2)². The graphs G_d are the currently largest known vertex-transitive graphs of diameter 2 and given degree d. They were originally constructed as lifts of the graphs K^*_q,q obtained from the complete bipartite graphs K_q,q by attaching (q-1)/4 loops to each vertex. Recently, the author simplified the construction of the graphs G_d by showing that they can be obtained as regular covers of the graphs D^*_q, the q -dipoles with (q-1)/4 loops at each of the two vertices In this contribution we show that certain automorphisms of the graphs G_d are not lifts of the automorphisms of D^*_q.

Keywords: regular coverings of graphs; voltage assignments; lifts of automorphisms

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