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 creation to the examine literature and to big open questions in smooth algebraic graph theory"—L'ENSEIGNEMENT MATHEMATIQUE
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Extra info for Algebraic Graph Theory
R. A weak path is a sequence uo , . . , Ur of distinct vertices such that for i = 1 , . . , r, either ( ui- I . lli ) or ( u i , Ui J ) is an arc. ) A directed graph is strongly connected if any two vertices can be joined by a path and is weakly connected if any two vertices can be joined by a weak path. A directed graph is weakly connected if and only if its "underlying" undirected graph is connected. (This is often used as a definition of weak connectivity. ) A strong component of a directed graph is an induced subgraph that is maximal, subject to being strongly connected.
If H and K are subgroups and g E G , then HgK is called a double coset . The double coset HgK is a union of right cosets of H and a union of left cosets of K, and G is partitioned by the distinct double cosets HgK, as g varies over the elements of G. : G. Show that each orbit of H on V corresponds to a double coset of the form GxgH. Also show that the orbit of Gx corresponding to the double coset GxgGx is self-paired if and only if GxgGx = Gxg - 1 Gx . 1 1 . Let G be a transitive permutation group on V .
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. 15. Show that if X is a graph with minimum valency at least four, Aut ( X) and Aut(L(X)) are isomorphic. 16. Let S be a set of nonzero vectors from an m-dil1ensional vector space. Let X(S) be the graph with the elements of S as its vertices, with two vectors x and y adjacent if and only if xry # . 0. ) Show that any independent set in X(S) has cardinality at most m .