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 obtainable advent to the examine literature and to big open questions in sleek algebraic graph theory"—L'ENSEIGNEMENT MATHEMATIQUE

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Xr } . If S is a subset of V, then the stabilizer Gs of S is the set of all permutations g such that Sg S. Because here we are not insisting that every element of S be fixed this is sometimes called the setwise stabilizer of S. If S = {xi > . . , Xr } , then Gx1 , ... ,x r is a subgroup of Gs . 1 Let G be a permutation group acting on V and let S be an orbit of G. 2. Counting 21 map x to y is a right coset of Gx . Conversely, all elements in a right coset of Gx map x to the same point in S.

We content ourselves with proving the following weaker result. Lemma 1 . � c that X and Y are graphs with minimum valency jour. Then X �Y if and only if L(X) � L Y ( ). Proof. Let C be a clique in L(X) containing exactly c vertices. If c > 3, then the vertices of C correspond to a set of c edges in X, meeting at a common vertex. Consequently, there is a bijection between the vertices of X and the maximal cliques of L(X) that takes adjacent vertices to pairs 1 . Graphs 12 of cliques w i th a vertex in common.

Let T(X) be the graph with the span­ ning tre of X as its vertices, where two spanning trees are adjacent if the symmetric difference of their edge sets has size two. Show that T( X) is connected. Show that if two trees have isomorphic line graphs, they are isomorphic. Use Euler's identity to show that K5 is not planar. Construct an infinite family of self-dual planar graphs. 24. A graph is self-complementary if it is isomorphic to its complement. Show that L(K3,3 ) is self-complementary. 25. Show that if there is a self-complementary graph X on n vertices, then n = 0, 1 mod 4.

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