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

By W.D. Wallis

Concisely written, light advent to graph concept appropriate as a textbook or for self-study

Graph-theoretic functions from different fields (computer technology, engineering, chemistry, administration science)

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

Many extra adjustments, advancements, and corrections caused by lecture room use

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Extra info for A Beginner’s Guide to Graph Theory

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For example, if a graph represents a road system, a common weight is the length of the corresponding stretch of road. In the underlying graph of a multigraph, w (x, y) might be the multiplicity of (x, y) in the multigraph. Weights also often represent costs or durations. The weight of a path P is the sum of the weights of the edges in P. Similarly, one can define the weights of walks, cycles, subgraphs and graphs. 2. Example. The routes traveled by an airline can conveniently be shown in a graph, with vertices representing cities and edges representing services.

Lt is clear that K'(G) ~ o(G), because one can disconnect G by removing all edges incident with any one given vertex. 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.

Since y has degree at least k in G, and H contains only K- 1 edges, there must be at least one edge adjacent to y, say yz, which is not an edge of H. Then H + y z is isomorphic to T. 1 Show that there are exactly six nonisomorphic trees on six vertices. 1. 3 Prove that a finite graph on v vertices that contains no cycle is connected if and only if it has v - 1 edges. 4 Prove that a connected graph is a tree if and only if it has the following property: If x and y are distinct vertices, then there is a unique path in G from x to y.

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