By Chris Godsil, Gordon F. Royle
C. Godsil and G.F. Royle
Algebraic Graph Theory
"A welcome boost to the literature . . . fantastically written and wide-ranging in its coverage."—MATHEMATICAL REVIEWS
"An available creation to the examine literature and to special open questions in sleek algebraic graph theory"—L'ENSEIGNEMENT MATHEMATIQUE
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Extra resources for Algebraic Graph Theory
1 7. Show that Aut(J(4n - 1 , 2n- 1 , n - 1 ) ) contains a subgroup isomorphic to Sym(4n). Show further that w(J(4n - 1 , 2n - l , n - 1 ) ) ::; 4n - 1 , and characterize the cases where equality holds. Notes The standard reference for permutation groups is Wielandt's clasic [5) . We also highly recommend Cameron  . For combinatorialists these are the hest starting points. However, almost every book on finite group theory contains enough information on permutation groups to cover our modest needs.
For any g E Sym(V), let orb2(g) denote the number of orbits of g in its action on E(Kv ) Then Burnside's lemma yields that the number of isomorphism clas of graphs with vertex set V is equal to . n. L 2orb2(9) . gESym (V) (2. 1) If all graphs were asymmetric, then every isomorphism class would contain n! graphs and there would be exactly 2m n! isomorphism clas. Our next result shows that in fact, the number of isomorphism clas of graphs on n vertices is quite close to this, and we will deduce from this that almost all graphs are asymmetric.
Proof. Suppose that the proportion of isomorphism clas of graphs on V that are asymmetric is p,. / 2 graphs, whence the average size of an isomorphism class is at most ( n! p, + ( 1 - p,) ) = n! 2 (1 + ,w) . Consequently, n! 2 (;) 2 (1 + p,) (1 + o(l)) n! > 2 (•) 2 , 2 . 4 . Orbits on Pairs 25 from which it follows that 11 tends to 1 as n tends to infinity. ), it follows that the proportion of graphs on D n vertices that are asymmetric goes to 1 as n tends to oo. Although the last result assures us that most graphs are asymmetric, it is surprisingly difficult to find examples of graphs that are obviously asymmetric.