By Rene J. Francillon
Profile guides quantity a hundred and sixty: The Mitsubishi G3M "Nell" by way of Rene J. Francillon, Ph.D. 14 pages.
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Then G − W contains at least two components and so it has a component D such that d := |V (D)| ≤ n−k . 14. 14: Estimate for degG (x). Then x can be adjacent to at most d − 1 vertices in G − W and k vertices in W . So, degG (x) ≤ d − 1 + k ≤ n−k 2 −1+k = n+k−2 , 2 which is a contradiction to the hypothesis. If G is simple and diam(G) = 1, then G = Kn and so k1 (G) = n − 1 = δ(G). The next results shows that simple graphs with diameter 2 also attain maximum possible edge-connectivity. 22 (J. Plesink, 1975).
If akj = 0, that is, if vk and vj are not adjacent. By induction, [Ar ]ik is the number of walks of length r connecting vi and vk . Each of these walks is extendable to a walk of length r + 1 connecting vi and vj iff vk and vj are adjacent, that is, iff akj = 1. So, by the above equations, [Ar+1 ]ij is the number of walks of length r + 1 connecting vi and vj . 12. If G is a connected simple graph, then the distance between vi and vj is the smallest integer p(≥ 0) such that [Ap ]ij = 0. Proof. By the minimality of p, [Ar ]ij = 0, for every r, 0 ≤ r ≤ p − 1.
Basic step: d(x, y) = 1. Let e be an edge joining x and y. Then G − e is connected; else, x or y is a cut-vertex which is a contradiction, since G is a block. Hence, there exists a path P (x, y) in G−e. But then (x, e, y) and P (x, y) are two internally disjoint (x, y)-paths. Induction step: d(x, y) > 1. Let P (x, y) be a shortest (x, y)-path; so length of P = d(x, y). Let w be a vertex that precedes y in P . So, d(x, w) = d(x, y) − 1. 11. 11: Construction of two internally disjoint (x, y)-paths in G.