Chris Godsil, Gordon F. Royle's Algebraic Graph Theory PDF

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 . . . fantastically written and wide-ranging in its coverage."—MATHEMATICAL REVIEWS

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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 .

Therefore , t) . 2. 13 it follows l. Then each vertex in A 3 . 4 . Vertex Since X e < k, and so if I A I 2: k, then l oA I ;: k. IAI < k. Since e s I AI 1 , it follows that is connected, may asume that ioAi 2: I AI ( k + 1 k. Therefore, Hence we IAI = 1 or k, when it is in all cases. o Vertex Connectivity verlex cutset A ioA i 2: k 39 IAI). 4 Connectivity X in a graph is a set of vertices whose deletion increases the number of connected components of X . The connectivity) of a connected graph X in a vertex cutset, and will be denoted by say that X is k-connected.

Therefore, g E H, and because B is an orbit of H we have B = B9 . 6. Primitivity and Connectivity 29 and because H < G, it does not consist of the whole of V , and hence it is 0 a nontrivial block of imprimitivity. 6 Primitivity and Connectivity Our second characterization of primitive permutation groups uses the orbits of G on V x V, and requires some preparation. A path in a directed graph D is a sequence uo , . . , Ur of distinct vertices such that ( Ui l , Ui) is an arc of D for i = 1 , . .

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Algebraic Graph Theory by Chris Godsil, Gordon F. Royle


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