Classification of 2-arc-transitive dihedrants

• Authors: Shaofei Du;Aleksander Malnič;Dragan Marušič
• Venue: Journal of Combinatorial Theory Series B
• Year: 2008

A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marusic, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. The list consists of the following graphs:(i)cycles C"2"n, n=3; (ii)complete graphs K"2"n, n=3; (iii)complete bipartite graphs K"n","n, n=3; (iv)complete bipartite graphs minus a matching K"n","n-nK"2, n=3; (v)incidence and nonincidence graphs B(H"1"1) and B^'(H"1"1) of the Hadamard design on 11 points; (vi)incidence and nonincidence graphs B(PG(d,q)) and B^'(PG(d,q)), with d=2 and q a prime power, of projective spaces; (vii)and an infinite family of regular Z"d-covers K"q"+"1^2^d of K"q"+"1","q"+"1-(q+1)K"2, where q=3 is an odd prime power and d is a divisor of q-12 and q-1, respectively, depending on whether q=1(mod4) or q=3(mod4), obtained by identifying the vertex set of the base graph with two copies of the projective line PG(1,q), where the missing matching consists of all pairs of the form [i,i^'], i@?PG(1,q), and the edge [i,j^'] carries trivial voltage if i=~ or j=~, and carries voltage h@?@?Z"d, the residue class of h@?Z, if and only if i-j=@q^h, where @q generates the multiplicative group F"q^* of the Galois field F"q.