Download A Beginner’s Guide to Graph Theory by W.D. Wallis PDF

By W.D. Wallis

ISBN-10: 1475731361

ISBN-13: 9781475731361

Concisely written, light creation to graph conception compatible as a textbook or for self-study

Graph-theoretic purposes from various fields (computer technology, engineering, chemistry, administration science)

2nd ed. contains new chapters on labeling and communications networks and small worlds, in addition to elevated beginner's material

Many extra adjustments, advancements, and corrections because of school room use

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Extra resources for A Beginner’s Guide to Graph Theory

Sample text

Since G has at least three vertices and no cutpoint, it contains no bridge. So every vertex adjacent to x is in X, and X is not empty. Assurne y is not in X; weshall derive a contradiction. Select a vertex z in X suchthat the distance d(y, z) is minimal; Iet Po be a shortest y-z path, and write Pt and P2 for the two disjoint x-z paths that make up a cycle containing x and z. 6); say Q is such a path. Let b be the vertex nearest to x in Q that is also in Po, and a the last vertex in the x-b section of Q that lies in Pt U P2; without loss of generality we can assume a is in Pt.

Suppose T = [X. Y] is a cutset of minimal size in G, where X U Y = V (G) andX n Y = 0. Then K'(G) = ITI. ), where V = IV(G)I. , andin this case the theorem is easily seen to be true. 3 Connectivity 41 So Iet us assume that there exist vertices x e X and y e Y that are not adjacent. Define S {p: p E Y, px E T} U {q: q EX, q # x, qy E T}. = Then G - S is a subgraph of G - T. Both x and y are vertices of G - S, and they are in different components of G - T, so they are in different components of G- S, and G- S is not connected.

6 Prove that for every v there exists a graph on v vertices such that, for any two nonadjacent vertices x and y, d (x) + d (y) :::= v - 1. 7 G is a graph with v vertices; x and y are nonadjacent vertices of G satisfying d (x) + d (y) ::: v. Prove that G + u v is Hamiltonian if and only if G is Hamiltonian. 8 G is a graph with v vertices; v :::= 3. (i) Prove that if G has at least v 2 -~v+ 6 edges then G is Hamiltonian. (ii) Find a nonHamiltonian graph with v 2 -~v+ 6 - 1 edges (thus proving that the preceding result is best-possible).

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