1. Graph Theory: Cliques Ryan Nickerson Roy Burdin Henry Armstrong Alex Martin Derek Hanrahan

2. Basic Graph Theory Vocab. Vertices are points of a graph. Usually used to represent something, in our example our vertices will represent towns. Edges are lines that connect vertices. Edges show a connection between vertices, and in our example edges will represent roads. A Graph is a collection of vertices and edges, in our example the graph will represent a county. Cliques of a graph are defined as follows: A subset S of vertices satisfying the following properties: –S contains 3 or more vertices –Each pair of vertices in S have an edge connecting them –S is maximal (there is no larger set of vertices that satisfies the second property and contains S) In our project we will show how to find cliques of a graph.

3. Example of Cliques in a Graph In the graph to the left you can see that it contains 4 vertices and 4 edges interconnecting them. We can see that there is a clique containing points A, B, and C because there is a edge between each pairing of these points. The first and third properties of being a clique are usually quite obvious to observe, as it is with this case.

4. Towns in Maine Our Problem: There are twenty towns in a county in Maine. Below these towns are listed in pairs if there is a road connecting them. We want to know if there are cliques within our graph and what towns belong to them. Town Connections: AB, AM, AD, BL, BK, BE, BC, CE, DE, EJ, EL, EK, FJ, GH, GJ, HJ, HP, IO, IR, IS, IQ, KN, KL, LM, MR, OQ, OS, OR, PQ, PR, QR, QS, QT, RS, ST

5. Using Matrices to Find Cliques Now to detect if a graph has cliques, because that may not be apparent as it is with our problem, you can use an incedence matrix. To make an incedence matrix you make an n x n matrix (n is the number of vertices), and label your rows and columns with your vertices. You then fill in your entries by either putting a 1 or a 0 in, 1 if the vertices have an edge directly connecting them or 0 if they don’t.

6. Our Incedence Matrix

7. Matrix Multiplication We now need to cube our matrix, but first we’ll explain matrix multiplication. To multiply two n by n matrices you take the first row of your first matrix and multiple its entries by the entries of the first column of you second matrix and then add those products together to get your 11 entry of your product matrix. So in other words, to get the ij-th entry of you product matrix take the i-th row of you first matrix, multiply its entries by the entries of the j-th column of your second matrix, and then add all those products together.

8. Cubed Incedence Matrix

9. Do we have any Cliques? So how do we use this cubed matrix? Well, you look at your ii-th entries (i.e. the diagonal entries) and if their entry is a positive number then you know your i-th vertice belongs to a clique. This is because our cubed incedence matrix tells us the number of ways your i-th vertice has access to itself, and since there are no loops in our graph then that i-th vertice must be part of a clique.

10. Towns in a Clique To the right is a list of towns that belong to cliques. We know they belong to cliques because looking back at our cubed matrix we can see in each towns case, where its row intersects its column you have a positive number. Towns belonging to a clique: B, C, E, G, H, I, J, K, L, O, P, Q, R, S, T So now that we know what towns belong to a clique we want to know which towns are in cliques together. For this task we’ll take a look at a graph of our points.

11. Graph of our Towns Note: This is just one possibility for how our graph looks, it could have easily been drawn a different way as long as it has all the same points connected

12. Our Cliques So after analyzing our graph and already knowing what towns belong to cliques we can determine that the cliques of our county are: BCE, BEKL, GHJ, QST, PQR, and IOQRS. All 6 of these cliques also meet all three criterion for being a clique.

13. What does it all mean? So after determining what towns are in a clique in our county, what do we do with this information. Well, in this specific example we can say the towns that create cliques could be seen as ‘economic centers’ (perhaps not that level in Maine, but if you generalized the idea). Since the towns are all interconnected by roads you can easily transport goods and services within those towns. Cliques also have a variety of other uses, such as determining the validity of communication lines (making sure everyone can contact everyone else within a group). Cliques could also be used to determine degrees of separation between people. For example, on Facebook there is an experiment called Six Degrees of Separation whose purpose is to see if everyone on Facebook is only six people away from one another. If it were feasible to create a incedence matrix of everyone on Facebook you could us cliques to determine if everyone is only separated by six people.