Automorphisms of a Family of Cubic Graphs

Abstract

A Cayley graph Cay(G, S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G, S). For a positive integer n, let Γn be a graph with vertexIn this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to Z 3 2 S3 for n = 2, and to Z n 2 D2n for n > 2. Furthermore, we determine all pairs of G and S such that Γn = Cay(G, S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.2010 Mathematics Subject Classification: 05C25, 20B25

Trivalent dihedrants and bi-dihedrants

“…Motivated by the above mentioned facts, we shall focus on trivalent non-arc-transitive dihedrants. Our first theorem generalizes the results in [22,25] to all trivalent dihedrants. Recall that for an integer m ≥ 2, the cross ladder graph CL 4m has vertex set V 0 ∪ V 1 ∪ .…”

“…1 for CL 4m ). It is worth mentioning that the cross ladder graph plays an important role in the study of automorphisms of trivalent graphs (see, for example, [5,25,21]). Motivated by the above mentioned facts, we shall focus on trivalent non-arc-transitive dihedrants.…”

On the Normality of Cubic Cayley Graphs of Order 8p2

“…By Ref. [23], we have S 0 = b b −1 ba , S 1 = b b −1 ab and S 3 = b bc ba , relatively. If X P = Q 3 , we have Aut X P S 4 × 2 .…”

“…We assume that X is non-symmetric. Then the vertex-stabilizer [23]) for some positive integer n ≥ 2. Let P be a Sylow p-subgroup of A such that P ≤ R G .…”

Trivalent vertex-transitive bi-dihedrants

“…1 for CL 4m ). It is worth mentioning that the cross ladder graph plays an important role in the study of automorphisms of trivalent graphs (see, for example, [5,21,25]). Motivated by the above mentioned facts, we shall focus on trivalent non-arc-transitive dihedrants.…”