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 learn literature and to big open questions in smooth algebraic graph theory"*—L'ENSEIGNEMENT MATHEMATIQUE

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**Additional resources for Algebraic Graph Theory**

**Example text**

1 ) this implies that IN(A u B) I -s; From ( a ) and ( b ) we se that I A U Bl + IN(A U B) l < n. 3 ( d ) . (d ) IN(A u B) I = K.

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. The remaining details are left exercise. There is as an 0 another interesting characterization of line graphs: Theorem 1 .

23. n vertices has n - 1 edges. Let X be a connected graph. 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.

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