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

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5. 4 Let X be a graph on n vertices with connectivity K . Sup pose A and B are fragments of X and A n B ::/: 0. If I A I $ I B I , then A n B is a fragment. Proof. 5. The cardinalities of five of these pieces are also defined in this figure. We present the proof as a number of steps. ( a) IA U B l < Since n - IFI + IFI and therefore = K. n - K for any fragment IAI + IBI $ ( b) IN(A u B) I $ K . From Lemma c + d + e. 3 we = n - K . Since find that n - of K - X, I BI , A n B is nonempty, I N(A n B) I $ a + b + c the claim follows.

Therefore, J(v, k, with valency i) has m vertices, and it is a regular graph As the next result shows, we can asume that v � 2k. 1 If v � k � i, then J (v , k, i) � J( v , v - k , v - 2 k + i). Proof. The function that maps a k-set to i ts complement in n i s an iso morphism from J ( v , k, the details. 2k, i) 2 to J(v, v - k , v - k + i); you are invited to check D For v � the graphs J(v, k, k - 1 ) are known as the Johnson graphs, and the graphs J (v , k, 0) are known as the Kneser graphs, which we will study in some depth in Chapter 7.

Combin. Theory Ser. B , 5 3 (1991), 4Q-79. H. WIELAN DT, Finite Pe rmutation Groups, Academic Press, New York, 1964. 3 Transitive Graphs We are going to study the properties of graphs whose automorphism group acts vertex transitively. A vertex-transitive graph is necesarily regular. One challenge is to find properties of vertex-transitive graphs that are not shared by all regular graphs. We will se that transitive graphs are more strongly connected than regular graphs in general. Cayley graphs form an important clas of vertex-transitive graphs; we introduce them and offer some reasons why they are important and interesting.