Applications of Linear Algebra to Graph Theory MATH Cutler

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1 Applications of Linear Algebra to Graph Theory MATH -00 Cutler Introduction Graph theory is a relatively new branch of mathematics which deals with the study of objects named graphs. These types of graphs are not of the variety with an x- and y-axis, but rather are made up of vertices, usually represented as points, and edges, usually thought of as lines in between two vertices. For example, the following are examples of graphs: P K K, P, called the path on vertices, can be generalized in the obvious way to a path on n vertices, P n. Likewise, K n is the complete graph on n vertices, in which all possible edges are present (NB: we do not consider graphs with more than one edge between two vertices). Lastly, K r,s is the complete bipartite graph with parts made up of r and s vertices. In general, a bipartite graph is a graph in which the vertices can be split into two parts, where all edges of the graph are in between these two parts. That is, the two parts have no edges inside of them. Note that a bipartite graph need not be a complete bipartite graph. We may label the vertices of our graph in any way we would like, but, for example, we could label the vertices of the graphs above in the following ways: P K K, 6

2 Armed with these labellings, we are ready to see a matrix that can be associated with each of these graphs. Before we do this, let us discuss a bit of notation. Now, we simply refer to a vertex in a graph by its label. Thus, we call the leftmost vertex in the representation above of P simply. Further, if two vertices i and j are adjacent, or have an edge between them, we write i j. Likewise, if they are not adjacent, or do not have an edge between them, we write i j. So, in P as labelled above,,, but. Also, note that if i j then j i. We define the adjacency matrix of a graph G (we assume that G has n vertices) to be the n n matrix A(G) = (a ij ) with { if i j a ij = 0 if i j Thus, we can see that the adjacency matrices of the graphs as labelled above are: A(P ) = A(K ) = A(K, ) = You may have already noticed some characteristics of these matrices. First of all, we have (implicitly up to this point) assumed that there are no loops, or edges which go from a vertex to itself. Thus, for any such graph, there are zeros down the diagonal entries, or a ii = 0 for i =,...,n. Also, since each vertex is represented by both a row and a column, the adjacency matrix is symmetric, or has a ij = a ji for all j =,...,n and i =,...,n. Powers of the adjacency matrix One application of the adjacency matrix to graph theory is found by taking powers of the adjacency matrix. Let us examine what happens when we let A = A(P ). Then, of course, A = A. Further, A = = 0 0 and A = =

3 You may be able to see a pattern emerging. Before we start congratulating ourselves, let us try another small graph. Let B = A(K ), where K is the complete graph on vertices. So and B = B = B = , = = As it turns out, there is a very nice relationship between powers of the adjacency matrix and a basic characteristic in graphs. A walk in a graph G is a sequence of vertices where every two consecutive vertices have an edge between them. So, for example, in our labelled copy of P above, is a walk which goes back and forth across the entire path. The length of a walk is one less than the number of vertices in the walk (it is the number of edges traversed on the walk). So the length of is. Now we are ready to state the theorem which gives the fundamental relationship between adjacency matrices and walks in graphs. Theorem If A = (a ij ) is the adjacency matrix of a graph G and we let A r = (a (r) ij ), i.e., a(r) ij is the entry in the ith row and jth column of A r, then the number of walks of length r between vertex i and vertex j in G is exactly a (r) ij. Let us consider A = A(P ), as above. Then, of course, the entries of A = A give the number of walks of length between any two vertices as a walk of length is simply an edge. As for A, a = a = since we only have the walks and for walks of length starting and ending at and, respectively. But for walks of length two starting and ending at, we have both and, and thus a =. The only other walks of length two are and, and thus a = a =. The other entries are therefore 0. We shall leave you to ponder B on your own. Eigenvalues of the adjacency matrix Another set of natural objects to study are the eigenvalues of the adjacency matrix. Let us find the eigenvalues of A = A(P ). We know that λ is an.

4 eigenvalue of A if and only if det(a λi) = 0. Thus λ 0 det(a λi) = λ 0 λ = λ λ λ 0 λ + 0 = λ(λ ) ( λ) = λ( λ + ) So, λ = 0 or ±. Thus the eigenvalues of A are 0, and. Maple can be used to find eigenvalues of larger matrices must faster (and more accurately) than me. However, let us write out one more example, that of B = A(K ). Then det(b λi) = = λ λ λ λ λ λ λ + λ = λ(λ ) ( λ ) + ( + λ) = λ + λ + = (λ )(λ + )(λ + ) = 0 Thus, the eigenvalues of B are, and. The next natural question to ask is what these eigenvalues have to do with the original graph. In fact, they give quite a bit of information about the graph, but we shall be interested in only one bit. The degree of a vertex i in G is the number of edges which have i as an endpoint. Thus, in P, the degree of is, is and is again. In K, all vertices have degree. A graph with all degrees the same is called a regular graph. Thus, both K and K, are regular graphs. One last bit of notation, the maximum degree of a vertex of a graph G is denoted (G). Thus, (P ) =. Also, (K ) = and (K, ) =. We are now ready to state the main theorem of this section: Theorem If λ max (G) is the the maximum value of an eigenvalue of the adjacency matrix of G, then λ max (G) (G). The theorem does indeed hold true for our two examples, as λ max (P ) = = (P ) and λ max (K ) = = (K ). You will investigate this relationship a bit further in the exercises.

5 Exercises Directions: These exercises are meant to be a minimum requirement for the project. Additional work is encouraged. Form groups of two or three and use Maple to answer the questions below. You must describe your answers as well as submitting a Maple worksheet.. Write out the adjacency matrices of P, K 6 and K,. Make a conjecture about what the adjacency matrices of P n, K n and K r,s look like.. Use Maple to take higher powers of A(P ) and A(K ). Can you see a pattern emerging? Make a conjecture about the general form of powers of these adjacency matrices. Repeat this process for A(P 6 ), A(K ) and A(K, ). Can you make general conjectures about the general form of powers of A(P n ), A(K n ) and K r,r? What does this say about walks between different vertices of these graphs? Try to categorize the vertices of each type of graph according to the walks of different lengths originating at each vertex.. A cycle of length n, denoted C n, is a graph formed from P n by connecting the two end vertices. For example, C 6 is shown below. Find A(C n ) for C 6 n =,,, 6. Try to find a general form for A(C n ). Also, find the eigenvalues of A(C n ) for n =,,, 6. Describe your findings. Can you account for the differences? Make a conjecture about the eigenvalues of C n for general n. What is the difference between odd and even cycles? Is there any similarity between odd or even cycles and K r,s?. Use Maple to find the eigenvalues of A(P ), A(K 6 ) and A(K, ). For which of these graphs does λ max (G) = (G)? What about for the cycles in question? Make a conjecture about what graphs have λ max (G) = (G).