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Download e-book for iPad: Algorithmic Graph Theory by David Joyner, Minh Van Nguyen, Nathann Cohen

By David Joyner, Minh Van Nguyen, Nathann Cohen

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Xn } and let the edge set be E = {Xi Xi+1 | 0 ≤ i < n}. The resulting graph G = (V, E) is called the linear congruential graph of the linear congruential sequence S. See chapter 3 of Knuth [119] for other techniques for generating “random” numbers. (a) Compute the linear congruential sequences Si with the following parameters: (i) (ii) (iii) (iv) S1 : S2 : S3 : S4 : m = 10, m = 10, m = 10, m = 10, a = c = X0 = 7 a = 5, c = 7, X0 = 0 a = 3, c = 7, X0 = 2 a = 2, c = 5, X0 = 3 (b) Let Gi be the linear congruential graph of Si .

We also write the complement of G as G. The sum of the adjacency matrix of G and that of Gc is the matrix with 1’s everywhere, except for 0’s on the main diagonal. A simple graph that is isomorphic to its complement is called a selfcomplementary graph. Let H be a subgraph of G. The relative complement of G and H is the edge deletion subgraph G − E(H). That is, we delete from G all edges in H. Sage can compute edge complements, as the following example shows. sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : Gc = G .

That is, we delete from G all edges in H. Sage can compute edge complements, as the following example shows. sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : Gc = G . complement () sage : EG = Set ( G . edges ( labels = False )); EG {(1 , 2) , (4 , 5) , (1 , 4) , (2 , 3) , (3 , 6) , (1 , 6) , (2 , 5) , (3 , 4) , (2 , 4)} sage : EGc = Set ( Gc . edges ( labels = False )); EGc {(1 , 5) , (2 , 6) , (4 , 6) , (1 , 3) , (5 , 6) , (3 , 5)} sage : EG . difference ( EGc ) == EG True sage : EGc .

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Algorithmic Graph Theory by David Joyner, Minh Van Nguyen, Nathann Cohen


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