By W.D. Wallis
Concisely written, mild advent to graph thought appropriate as a textbook or for self-study
Graph-theoretic functions from assorted fields (computer technology, engineering, chemistry, administration science)
2nd ed. comprises new chapters on labeling and communications networks and small worlds, in addition to accelerated beginner's material
Many extra adjustments, advancements, and corrections as a result of school room use
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Additional info for A Beginner’s Guide to Graph Theory
So the algorithm finds route EMNSE, with cost $520. A different result is achieved if one starts at Nashville. Then the first edge selected is NE, with cost $110. The next choice is EM, then MS, then SN, and the resulting cycle NEMSN costs $530. To apply the sorted edges algorithm, first sort the edges in order of increasing cost: EM($100), EN($110), ES($120), MN($130), MS($150), NS($170). Edge EM is included, and so is EN. The next choice would be ES, but this is not allowed because its inclusion would give degree 3 to E.
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. 3(b). We now construct two x-b paths RandS. To form R, follow Pt from x to a and Q from a tob.
The weighted distance function W (x, y) has the following properties: (i) W (x, y) = 0 if and only if x = y; (ii) W(x, y) = W(y, x); (iii) W(x, y) + W(y, z) ~ W(x, z) for a/1 vertices x, y, z in G. The proof is left as an exercise. 8. In many applications it is desirable to know the path of least weight between two vertices. This is usually called the shortest path problem, primarily because a common application is one in which weights represent physical distances. We 20 2. Walks, Paths and Cycles shall describe an algorithm due to Dijkstra  that finds the shortest path from vertex s to vertex t in a finite connected graph G.
A Beginner’s Guide to Graph Theory by W.D. Wallis