By Chris Godsil, Gordon F. Royle

ISBN-10: 0387952209

ISBN-13: 9780387952208

ISBN-10: 0387952411

ISBN-13: 9780387952413

C. Godsil and G.F. Royle

*Algebraic Graph Theory*

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

"*An obtainable creation to the study literature and to special open questions in sleek algebraic graph theory"*—L'ENSEIGNEMENT MATHEMATIQUE

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**Additional info for Algebraic Graph Theory**

**Example text**

Proof. Assume X. 11: = ��:1 {X) and let A and B be two distinct edge atoms in A U B = V(X) , then, since neither A nor B contains more than half vertices of X, it follows that If the I AI and hence that of V(X) . = IB I = � I V(X) I A n B = 0. So we may assume that AUB is a proper subset Now, the previous lemma yields I a( A U B) I + I a( A n B) I � 2��:, and, since A U B =/= V(X) and A n B =/= 0, this implies that I a( A U B) I = Ia( A n B) I = Since AnB 11:. is a nonempty proper subset of the edge atom impossible.

12. Show that the only primitive permutation group on V that contains a transposition is Sym ( V ) . 13. Let X be a graph such that Aut ( X ) acts transitively on V ( X ) and let B be a block of imprimitivity for Aut ( X ) . Show that the subgraph of X induced by B is regular. 14. Let G be a generously transitive permutation group on V and let B be a block for G. Show that G t B and the permutation group induced by G on the translates of B are both generously transitive. 32 References 15. Let G be a transitive permutation group on V such that for each element v in V there is an element of G with order two that has v as its only fixed point.

We will leave some details to the reader. The support of a per mutation is the set of points that it does not fix. We claim that among all permutations g E Sym(V) with support of size an even integer 2r, the maximum value of orb2 (g) is realized by the permutation with exactly r cycles of length 2. Suppose g E Sym(V) is such a permutation with r cycles of length two and n - 2r fixed points. Since g2 = e, all its orbits on pairs of elements from V have length one or two. There are two ways in which an edge {x, y} E E(Kv ) can be not fixed by g.

### Algebraic Graph Theory by Chris Godsil, Gordon F. Royle

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