Algebraic Graph Theory by Chris Godsil, Gordon F. Royle

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) .

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