Algebraic combinatorics by Christopher D. Godsil

By Christopher D. Godsil

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Many interesting properties in graph theory are dependent on certain sets of edges having even size. 1, arise when all vertices have even degree. 14. The vertices on the left represented people and the vertices on the right represented jobs. 14 A a B b C c D d E e 27 that a certain person can perform a certain job. There can never be an edge between two vertices on the left or between two vertices on the right. Such a graph is called a bipartite graph. Formally, a graph G is bipartite if its vertices can be partitioned into two sets V1 and V2 and every edge joins a vertex in V1 with a vertex in V2 .

3 Edge Counting 25 Example 1: Use of Theorem 1 Suppose we want to construct a graph with 20 edges and have every vertex of degree 4. How many vertices must the graph have? Let v denote the number of vertices. The sum of the degrees of the vertices will be 4v, and by the theorem this sum must be twice the number of edges: 4v = 2 × 20 = 40. Hence v = 10. Example 2: Edges in a Complete Graph How many edges are there in K n , a complete graph on n vertices? Recall that K n has an edge between all possible pairs of vertices.

A) Show that every circuit in the graph in Exercise 3(l) has at least five edges. (b) Use part (a) and the result of Exercise 20 to show that this graph is nonplanar. 22. (a) Give an example of a graph with regions consisting solely of squares (regions bounded by four edges) and hexagons, and with vertices of degree at least 3.

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