By Chris Godsil, Gordon F. Royle
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 advent to the learn literature and to big open questions in smooth algebraic graph theory"—L'ENSEIGNEMENT MATHEMATIQUE
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Extra info for Algebraic Graph Theory
3 Asymmetric Graphs A graph is asymmetric if its automorphism group is the identity group. e. , the proportion of graphs on n vertices that are asymmetric goes to 1 as n -+ oo . Our main tool will be Burnside's lemma. Let V be a set of size n and consider all the distinct graphs with vertex set V. If we let Kv denote a fixed copy of the complete graph on the vertex set V, then there is a one-to-one correspondence between graphs with vertex set V and subsets of E(Kv ) . Since Kv has (�) edges, the total number of different graphs is Given a graph X, the set of graphs isomorphic to X is called the iso morphism class of X .
The diameter of a graph is the maximum distance between two dis tinct vertices. ) Detem1ine the diameter of J ( v, k, k 1) when v > 2 k . - 9. Show that Aut(Kn) is not isomorphic to Aut (L(Kn ) ) if and only if n = 2 or 4. 10. Show that the graph Ks \ e (obtained by deleting any edge e from Ks) is not a line graph. 1 1 . Show that K1 ,3 is not an induced subgraph of a line graph. 12. Prove that any induced subgraph of a line graph is a line graph. 1 3 . 2 ) . 14. Find all graphs G such that L(G) � G.
13. An embedding of K6 in the projective plane The torus is an orientable surface, which can be represented physically in Euclidean 3-space by the surface of a torus, or doughnut. It can be represented on paper by a rectangle where the points on the bottom side are identified with the points directly above them on the top side, and the points of the left side are identified with the points directly to the right of them on the right side. The complete graph K7 is not planar, nor can it be embedded in the projective plane, but it can be embedded in the torus as shown in Figure 1.
Algebraic Graph Theory by Chris Godsil, Gordon F. Royle