Se p 20 06 Upper Bounds on the Automorphism Group of a Graph Discrete Mathematics 256 ( 2002 ) 489 - 493 . Ilia

We give upper bounds on the order of the automorphism group of a simple graph In this note we present some upper bounds on the order of the automorphism group of a graph, which is assumed to be simple, having no loops or multiple edges. Somewhat surprisingly, we did not find such bounds in the literature and the goal of this paper is to fill this gap. As a matter of fact, implicitly such bounds were contained in works dealing with the edge reconstruction conjecture and are the corollaries of a simple theorem which is presented below (Theorem 1). Therefore we bring together a few results spread in different, sometimes in difficult to reach, sources (see Theorem 2 below). In Theorem 3 we derive a new bound, based on the notion of a greedy spanning tree . This new bound improves, in many cases, the bounds (1) and (2) of Theorem 2. We will use the following notation. Let F be a spanning subgraph of a fixed copy of a graphG. The number of embeddings of F inG, that is the number of labeled copies of F in G, is denoted by |F → G|. Clearly |F → G| = s(F → G)aut(F ), where s(F → G) is the number of subgraphs of G isomorphic to F and aut(F ) is the order of the automorphism group of F . We also use n = n(G) for the number of vertices and e = e(G) for the number of edges of G. As usual, ∆G, δG and dG stand for the maximum, the minimum and the average degree of G respectively. The degree of a vertex v∈G is denoted by dG(v). Theorem 1 Let F be a spanning subgraph of a graph G, Then aut(G) ≤ |F → G| = s(F → G)aut(F ). Proof. Let φ : G → G be an automorphism of G and let F1 be a fixed copy of F in G. Then, as F is a spanning subgraph of G, φ is completely determined by the knowledge of φ(F1). Since the number of different images φ(F1) does not exceed |F → G|, the result follows. Some relevant estimates of |F → G|, s(F → G) and aut(F ) for graphs in general and for special families of graphs are known and have been obtained mainly in connection with the edge reconstruction conjecture. We try to collect them in the following Theorem 2 Let G be a connected graph, then aut(G) ≤ n(∆G)! (∆G − 1) n−∆G−1 (1) Let T be a spanning tree in G, then aut(G) ≤ ∆T ∆G (dG) n ∏ v ∈ V (G) (dT (v)− 1)! (2) Let p = p(G) be the path covering number of a graph, i.e. the minimum number of vertex-disjoint paths containing all vertices of G. Then aut(G) ≤ 2p n(26) (3) aut(G) ≤ (dG) ((∆G − 1)!) e−n+3−2δG (δG−1)(∆G−2) , (4)