By Chris Godsil, Gordon F. Royle

C. Godsil and G.F. Royle

*Algebraic Graph Theory*

*"A great addition to the literature . . . fantastically written and wide-ranging in its coverage.*"—MATHEMATICAL REVIEWS

"*An available advent to the learn literature and to special open questions in smooth algebraic graph theory"*—L'ENSEIGNEMENT MATHEMATIQUE

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Department of Computer Science, Australian National University, 1990. [6] D. B. WEST, Introduction to Graph Theory, Prentice Hall Inc. , Upper Saddle River, NJ, 1996. 2 Groups The automorphism group of a graph is very naturally viewed as a group of permutations of its vertices, and so we now present some basic informa tion about permutation groups. This includes some simple but very useful counting results, which we will use to show that the proportion of graphs on n vertices that have nontrivial automorphism group tends to zero as n tends to infinity.

Our next result shows that the stabilizers of two points in the same orbit of a group are conjugate. 3 Let G be a permutation group on the set V and let x be a point in V. If g E G, then g-1Gxg = Gx9 · Proof. Suppose that fixes y. Let h E xY = y. First we show that every element of g- 1 Gxg Gx. Then Y'q- 1 x hg xn y, and therefore g- 1 hg E Gy. On the other hand, if h E Gy , then ghg- 1 x, whence we se that g- 1 Gxg Gy . hg = = = = fixes o 22 2. Groups If g is a permutation of V, then fix( g ) denotes the set of points in V fixed by g.

This map ping is an automorphism because the k-tuples x and y differ in precisely one coordinate position if and only if x + v and y + v differ in precisely one coordinate position. There are 2k such permutations, and they form a subgroup H of the automorphism group of Qk . This subgroup acts transi tively on V(Qk ) because for any two vertices x and y, the automorphism 0 Py - x maps x to y. The group H of Lemma 3. 1 . 1 is not the full automorphism group of Qk . Any permutation of the k coordinate positions is an automorphism of Qk , and the set of all these permutations forms a subgroup K of Aut(Qk ) , iso morphic to Sym(k) .