Smallest graphs with given generalized quaternion automorphism group

Abstract

For n≥3, a smallest graph whose automorphism group is isomorphic to the generalized quaternion group is constructed. If n≠3, then such a graph has 2n+1 vertices and 2n+2 edges. In the special case when n=3, a smallest graph has 16 vertices but 44 edges.

Smallest graphs with given automorphism group

“…Suppose, conversely, that w ∼ a 2 u ⇐⇒ w ∼ a 6 u. (10) Let B = {w, a 4 w, u, a 4 u}. Define φ : V (Γ) → V (Γ) to be the map…”

“…In [4] it is shown that every finite group can be realised, up to isomorphism, as the automorphism group of a finite graph; in fact, for every finite group G there exist infinitely many finite graphs having automorphism group isomorphic to G. Given a finite group G, define α(G) to be the smallest number of vertices of any graph Γ having Aut(Γ) ∼ = G. The problem of finding α(G) has been considered by many authors. The value of α(G) has been determined in [1] for abelian groups G, in [8], [9], [11], [15] for dihedral groups G and in [10] for generalised quaternion groups G. The question has also been investigated for several families of finite simple groups in [14]. A recent survey on this problem can be found in [22].…”

An Efficient Structural Descriptor Sequence to Identify Graph Isomorphism and Graph Automorphism

“…On the dataset DS 5 Among the 18 graphs, we could identify 18 graphs to be non-isomorphic with distinct sequences, which has taken only 8.4866 seconds in total, the existing isomorphism algorithm on 18 2 pairs takes 8.3194 seconds. On the dataset DS 6 Among the 167 graphs, we could identify 146 graphs to be non-isomorphic with distinct sequences, and remaining 21 graphs were classified in to 8 equivalence classes, having v = {2, 2, 2, 2, 2, 3, 3, 5}, the elements of v are in each of the class.…”

“…Maximal clique and graph automorphism problems are interesting and long standing problems in graph theory. Lots research works are done on graph automorphism problem [14,15,16,17,18,19] and clique problem [20,21,22,25].…”

$16$-vertex graphs with automorphism groups $A_{4}$ and $A_{5}$ from icosahedron

“…There are groups which admit a graphical regular representation, for such groups µ(G) ≤ |G|. For some recent work see [6], [7], [9].…”