By Robin J. Wilson

Graph thought has lately emerged as a topic in its personal correct, in addition to being a massive mathematical device in such different topics as operational study, chemistry, sociology and genetics. This e-book presents a entire advent to the topic.

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9(v) to find the automorphism group of the Petersen graph. n rs n puzzles In this section we present three recreational puzzles that can be solved using ideas relating to graphs. I n each puzzle, note how the use of a graph diagram makes the prob­ lem much easier to understand. The eight circles problem Place the letters A, B, C, D, E, F, G, H into the eight circles in Fig. 1, in such a way that no letter is adjacent to a letter that is next to it in the alphabet. Fig. 1 First note that trying all the possibilities is not a practical proposition, as there are 8!

9 s 33 5 5 Let G be a connected graph. What can you say about (i) an edge of G that appears in every spanning tree? (ii) an edge of G that appears in no spanning tree? If G is a connected graph, a centre of G is a vertex v with the property that the maximum Counting trees 47 of the distances between v and the other vertices of G is as small as possible. By succes­ sively removing end-vertices, prove that every tree has either one centre or two adjacent centres. Give an example of a tree of each type with 7 vertices.

For example, i n the graph o f Fig. 3, the sets {e\, en,

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