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 welcome boost to the literature . . . superbly written and wide-ranging in its coverage."—MATHEMATICAL REVIEWS

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

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Example text

This proves ( c) . We leave as 41 not the an exercise. 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.

Construct a cubic planar graph on 1 2 vertices with trivial auto­ morphism group, and provide a proof that it has no nonidentity automorphism. 7. Decide whether the cube is a Halin graph. 8. Let X be a self-complementary graph with more than one vertex. Show that there is a permutation g of V ( X ) such that: ( a) {x, y} E E( X ) if and only if {x9, y9 } E E ( X ) , ( b ) fl E Aut( X ) but g2 =/= e, (c ) the orbits of g on V ( X ) induce self-complementary subgraphs of X. 9 . If G is a permutation group on V, show that the number of orbits of G on V x V is equal to 1 2 l fi x(g ) l GI I g EG and derive a similar formula for the number of orbits of G on the set of pairs of distinct elements from V.

Since each edge lies in two faces, we have 2e = 3 /, and H<> by Euler's formula, e = 3n - 6. A planar graph on n vertices with 3n - 6 edges is necesarily maximal; such graphs are called planar triangulations . 10 are planar triangulations. A planar graph can be embedded into the plane in infinitely many ways. The two embeddings of Figure 1 . 1 1 are easily sen to be combinatorially different : the first has faces of length 3 , 3, 4, a nd 6 while the second has faces of lengths 3, 3, 5, and 5. It is an important result of topological graph 14 1 .

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Algebraic Graph Theory by Chris Godsil, Gordon F. Royle

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