Graph Theory Lecture Notes 3

Application 1.3.5: The following committees need to have meetings scheduled. Are three meeting times sufficient to schedule the committees so that no member has to be at two meetings simultaneously? Why?

A = {Smith, Jones, Brown, Green}
B = {Jones, Wagner, Chase}
C = {Harris, Oliver}
D = {Harris, Jones, Mason}
E = {Oliver, Cummings, Larson}.
Ans: Yes Solution: We consider the committees as vertices of a graph. Join two vertices if the two committees have a common member (this corresponds to a conflict, the two committees would have to meet at different times). The question can then be rephrased as, can we assign one of three colors to each vertex so that no two adjacent vertices get the same color (The colors correspond to meeting times)? Since this graph contains a 3-cycle (a triangle) at least three colors are needed (one each for A, B and D), but then C and E can be colored with the colors used for A and B, so no more than 3 colors are needed.

Testing for Non-Isomorphism

Although showing that two graphs are isomorphic is in general very difficult, it is sometimes easy to show that two graphs are not isomorphic.

A numerical graph invariant is a numerical property of graphs for which any two isomorphic graphs must have the same value.

Some easy numerical graph invariants:

  1. The number of vertices.
  2. The number of edges.
  3. The number of components.
  4. The degree sequence.
  5. For any subgraph, the number of distinct copies of that subgraph.
For simple graphs, we also have a non-numerical invariant, namely the edge-complement of the graph. (Isomorphic simple graphs have isomorphic edge-complements.)

Example: Consider the two graphs:

Both graphs have 6 vertices, 8 edges and 1 component. The degree sequence of the left graph is <2,2,3,3,3,3> while that for the right graph is <2,2,2,3,3,4>, so they are not isomorphic.

Representing Graphs and Digraphs

Incidence Matrices: A matrix whose rows are indexed by the vertices and whose columns are indexed by the edges. The possible entries at row v and column e are:
For graphs
0 if v is not an endpoint of e
1 if v is an endpoint of e
2 if e is a self-loop at v
For digraphs
0 if v is not an endpoint of e
1 if v is the head of e
-1 if v is the tail of e
2 if e is a self-loop at v

Adjacency Matrices: A matrix whose rows and columns are indexed by the vertices in the same order. The possible entries at row u and column v are:
# edges between distinct u and v (for digraphs the # of arcs from u to v)
# self-loops at v

Proposition 2.5.4: Let G be a graph with adjacency matrix A. The (u,v) entry in Ar is the number of u-v walks of length r in G.

Arc Lists: A list which gives for each vertex the names of the out-arcs at the vertex (digraph), or the in-arcs at the vertex (digraph) or the edges incident with the vertex (graph).