Properties and Recent Applications in Spectral Graph Theory

Transcription

1 Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 8 Properties and Recent Applications in Spectral Graph Theory Michelle L. Rittenhouse Virginia Commonwealth University Follow this and additional works at: Part of the Physical Sciences and Mathematics Commons The Author Downloaded from This Thesis is rought to you for free and open access y the Graduate School at VCU Scholars Compass. It has een accepted for inclusion in Theses and Dissertations y an authorized administrator of VCU Scholars Compass. For more information, please contact licompass@vcu.edu.

2 Properties and Recent Applications in Spectral Graph Theory By Michelle L. Rittenhouse Bachelor of Science, University of Pittsurgh Johnstown, PA 989 Director: Dr. Ghidewon Aay Asmerom Associate Professor, Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, Virginia April, 8

3 To: David, Mom, Dad and Shannon. Thank you for all of you support and encouragement. I couldn t have done it without you. To: Nikki and Will. Through determination and perseverance, you can achieve anything.

4 Tale of Contents Introduction i Ch : Graph Theory Section.: Introduction to Graph Theory Section.: The Adjacency Matrix Section.: The Characteristic Polynomial and Eigenvalues Chapter : Spectral Graph Theory Section.: Introduction Section.: The Spectrum of a Graph Section.: Bipartite Graphs Section.4: Walks and the Spectrum Chapter : The Laplacian Section.: Introduction Section.: Spanning Trees Section.: An Alternative Approach Chapter 4: Applications Section 4.: Chemical Applications Section 4.: Protein Structures Section 4.: Graph Coloring Section 4.4: Miscellaneous Applications References

5 Astract There are numerous applications of mathematics, specifically spectral graph theory, within the sciences and many other fields. This paper is an exploration of recent applications of spectral graph theory, including the fields of chemistry, iology, and graph coloring. Topics such as the isomers of alkanes, the importance of eigenvalues in protein structures, and the aid that the spectra of a graph provides when coloring a graph are covered, as well as others. The key definitions and properties of graph theory are introduced. Important aspects of graphs, such as the walks and the adjacency matrix are explored. In addition, ipartite graphs are discussed along with properties that apply strictly to ipartite graphs. The main focus is on the characteristic polynomial and the eigenvalues that it produces, ecause most of the applications involve specific eigenvalues. For example, if isomers are organized according to their eigenvalues, a pattern comes to light. There is a parallel etween the size of the eigenvalue (in comparison to the other eigenvalues) and the maximum degree of the graph. The maximum degree of the graph tells us the most caron atoms attached to any given caron atom within the structure. The Laplacian matrix and many of its properties are discussed at length, including the classical Matrix Tree Theorem and Cayley s Tree Theorem. Also, an alternative approach to defining the Laplacian is explored and compared to the traditional Laplacian. i

6 CHAPTER : GRAPH THEORY SECTION.: INTRODUCTION TO GRAPH THEORY A graph G is a finite nonempty set of points called vertices, together with a set of unordered pairs of distinct vertices called edges. The set of edges may e empty. The degree of a vertex v, deg(v), is the numer of edges incident on v. A graph is regular if all vertices have equal degrees. A graph is considered a complete graph if each pair of vertices is joined y an edge. In the example graph elow, the set of vertices is V(G) = {a,, c, d} while the set of edges is E(G) = {a, ac, c, d, cd}. The graph is not complete ecause vertices a and d are not joined y an edge. deg(a) = = deg(d), deg() = = deg(c). Two vertices are adjacent if there is an edge that connects them. In Figure -, vertices a and c are adjacent, while vertices a and d are not. a Graph G Figure - c d A wide variety of real world applications can e modeled using vertices and edges of a graph. Examples include electrical nodes and the wires that connect them, the stops and rails of a suway system and communication systems etween cities. The cardinality of the vertex set V(G) is called the order of G and is denoted y p(g), or simply p when the context makes it clear. The cardinality of the edge set E(G) is called the size of G, and is usually denoted y q(g) or q. A graph G can e denoted as a G(p, q) graph.

7 A v i v j walk in G is a finite sequence of adjacent vertices that egins at vertex v i and ends at vertex v j. In Figure -, a walk from a to d would e acd. Another walk from a to d is acd. If each pair of vertices in a graph is joined y a walk, the graph is said to e connected. The distance etween any two vertices in a graph is the numer of edges traced on the shortest walk etween the two vertices. The distance from a to d is. The maximum distance etween all of the pairs of vertices of a graph is the diameter of the graph. In Figure -, the distance etween any two vertices is either or, making the diameter of G, or diam(g) =. The vertices and edges may have certain attriutes, such as color or weight. Also, the edges may have direction. When the edges are given direction we have a digraph, or directed graph, as shown elow in Figure -. Digraphs can e used to model road maps. The vertices represent landmarks, while the edges represent one-way or two-way streets. home ank work Figure - vcu gym An incidence matrix associated with a digraph G is a q p matrix whose rows represent the edges and columns represent the vertices. If an edge k starts at vertex i and ends at vertex j, then row k of the incidence matrix will have + in its (k, i) element, and it its (k, j) element. All other elements are.

8 a g a c d e h D = f f I = Figure - c e d g h Example: A directed graph and its incidence matrix are shown aove in Figure -. When two vertices are joined y more than one edge, like the example in Figure -4, it ecomes a multi-graph. a Figure -4 c d When a pair of vertices is not distinct, then there is a self-loop. A graph that admits multiple edges and loops is called a pseudograph. In the pseudograph elow, edge cc is joining a pair of non-distinct vertices; therefore, there is a self-loop at vertex c. a c d Figure -5 Graphs G and H are isomorphic, written G H, if there is a vertex ijection

9 f: V(G) V(H) such that for all u, v V(G), u and v adjacent in G f(u) and f(v) are adjacent in H. Likewise, u and v not adjacent in G f(u) and f(v) are not adjacent in H. a n G = H = m Figure -6 c d q p Graphs G and H are isomorphic, ecause a vertex ijection etween them is: f(a) = m, f() = n, f(c) = g, and f(d) = p. 4

10 SECTION.: ADJACENCY MATRICES There is a great deal of importance and application to representing a graph in matrix form. One of the key ways to do this is through the adjacency matrix, A. The rows and columns of an adjacency matrix represent the vertices, and the elements tell whether or not there is an edge etween any two vertices. Given any element, a ij, a ij = if a i and a j are connected otherwise Example: A graph G with its adjacency matrix A. G a c d A = a c d a c d Figure -7 Notice that the diagonal of an adjacency matrix of a graph contains only zeros ecause there are no self-loops. Rememer that our graphs have no multiple edges or loops. This causes the trace of the adjacency matrix, written tr(a), the sum of its main diagonal, to e zero. Also, when A represents a graph, it is square, symmetric, and all of the elements are non-negative. In other words, a ij = a ji. Property -: The numer of walks of length l from v i to v j in G is the element in position (i, j) of the matrix A l [Bi, p9]. Proof: If l =, then the numer of walks from v i to v j, i j, is, making element 5

11 (i, j) =. The numer of walks from any vertex to itself of length zero is, making element (i, i) =, giving us the identity matrix. A = I, therefore it checks if l =. If l =, then A = A, the adjacency matrix. Let Property - e true for l = L. The set of walks of length L+ from vertex v i to vertex v j is in ijective correspondence with the set of walks of length L from v i to v h adjacent to v j. In other words, if there is a walk of length L from v i to v h, and v h is adjacent to v j, then there is a walk of length L+ from v i to v j. The numer of such walks is ( A { vh, v j } E( G) L ) ih n L = ( A ) a = (A h= ih hj L+ ) ij Therefore, the numer of walks of length L+ joining v i to v j is (A L+ ) ij. By induction, the property holds. Example: Look again at graph G and its adjacency matrix A: G a c d A = a c d a c d Figure -7 The numer of walks from vertex to vertex d of length can e found y squaring matrix A. A =, therefore if we look at element (, d) = element (d, ) =. There is one walk from to d of length. That walk is cd. 6

12 4 Likewise, 4 A =. Element (a, c) = element (c, a) = 4. Therefore, there are walks of length from vertex a to vertex c. These walks are: acac, acdc, acc, aac. Property - has two immediate corollaries. We list them as Property - and Property -. Property -: The trace of A is twice the numer of edges in the graph. Proof: Let v i e an aritrary vertex of G. For every edge v i v j of G, y Property -, we get a count of towards the (i, j) position of A. That is, the (i, j) position of A is equal to the degree of v i, ecause every walk v i v i that involves only one v i v j edge will have a length of. This means: tr( A ) = v V ( G) i deg( v ) = q. Recall, q equals the cardinality of the i edge set of G. This can e seen in our example in Figure -7. A =. The trace of A is 8 and the numer of edges is 4. Property -: The trace of A is six times the numer of triangles in the graph. Proof: Let v i e an aritrary vertex of G. For every -cycle v i v j of G, again y Property -, we get a count of towards the (i, j) position of A. That is, the (i, j) position of A is equal to the numer of -cycles that start and end at v i. However, every -cycle v i v j v k v i is counted six times ecause the vertices of the -cycle can e ordered in 6 ways. This means that tr(a ) = 6(# of triangles) in the graph. 7

13 Again, let s test it with the graph from Figure -7. A =. The trace is 6 and the numer of triangles in the figure is, illustrating Properties - and -. 8

14 SECTION.: THE CHARACTERISTIC POLYNOMIAL AND EIGENVALUES The characteristic polynomial, Г, of a graph of order n is the determinant det(λi A), where I is the n n identity matrix. Example : given adjacency matrix A a c a = c a From graph G c Figure -8 Г = λ det( λi A) = det λ λ λ = det λ λ = λ λ The general form of any characteristic polynomial is λ n + c λ n + c λ n c n Equation Property -4: The coefficients of the characteristic polynomial that coincide with matrix A of a graph G have the following characteristics: ) c = ) c is the numer of edges of G ) c is twice the numer of triangles in G Proof: For each i {,,, n}, the numer ( ) i c i is the sum of those principal minors of A which have i rows and columns. So we can argue as follows: ) Since the diagonal elements of A are all zero, c =. 9

15 ) A principal minor with two rows and columns, and which has a non-zero entry, must e of the form. There is one such minor for each pair of adjacent vertices of the characteristic polynomial, and each has a value of. Hence, ( ) c = E(G). ) There are essentially three possiilities for non-trivial principal minors with three rows and columns:,,. Of these, the only non-zero one is the last one (whose value is two). This principal minor corresponds to three mutually adjacent vertices in Г, and so we have the required description of c. [Bi, p8-9] Example: This can e seen from the characteristic polynomial of A aove. n = 4 ) there is no λ, making c = ) c =. The numer of edges of G is 4. ) c =. The numer of triangles in G is. The characteristic polynomial is enormously important to spectral graph theory, ecause it is an algeraic construction that contains graphical information. It will e explored more in Chapter. The roots of a characteristic polynomial are called the eigenvalues. Setting the characteristic polynomial λ λ from Example equal to zero and solving gives us the eigenvalues {,, }. Eigenvalues are at the heart of understanding the properties and structure of a graph. We will also study them more in chapter. Property -5: The sum of the eigenvalues of a matrix equals its trace. Example: Let us consider a adjacency matrix

16 A = a c d. Det(A - λi) = λ (a + d)λ + (ad c). Applying the quadratic equation and setting = c, we get λ = ( a + d) + ( a d) + 4 and λ = ( a + d) ( a d) + 4. Adding a + d the eigenvalues gives us λ + λ = = a + d = the sum of the diagonal, or tr(a). Similar computations can e applied to any adjacency matrix, regardless of the dimensions. The algeraic multiplicity of an eigenvalue is the numer of times that the value occurs as a root of the characteristic polynomial. In example, the eigenvalue - has a multiplicity of and the eigenvalue of has a multiplicity of. The eigenvectors that correspond to the eigenvalues are. The geometric multiplicity is the dimension of the eigenspace, or the suspace spanned y all of the eigenvectors. Property -6: If a matrix is real symmetric, then each eigenvalue of the graph relating to that matrix is real. Proof: The proof follows from the Spectral Theorem from Linear Algera: Let A e a real, symmetric matrix. Then there exists an orthogonal matrix Q such that A = Q Λ Q - = Q Λ Q T, where Λ is a real diagonal matrix. The eigenvalues of A appear on the diagonal of Λ, while the columns of Q are the corresponding orthonormal eigenvectors. The entries of an adjacency matrix are real and the adjacency matrix is symmetric. Therefore, all of the eigenvalues of an adjacency matrix are real. [Ol, p49]

17 Property -7: The geometric and algeraic multiplicities of each eigenvalue of a real symmetric matrix are equal. Property -8: The eigenvectors that correspond to the distinct eigenvalues are orthogonal (mutually perpendicular). Note: When u and v are two orthogonal eigenvectors of A associated with two distinct eigenvalues λ and μ in that order, then the unit vectors u u v and v are orthonormal eigenvectors associated with λ and μ respectively. Property -9: If a graph is connected, the largest eigenvalue has multiplicity of. Let s check these properties with Example from aove: A a c a = c a From graph G c Figure -8 The eigenvalues are {,, }. All are real, demonstrating Property -6. The algeraic multiplicity of is two, and the algeraic multiplicity of is one. The eigenvector that corresponds to the eigenvalue - is the linear comination of the vectors,. We can easily show that the two vectors are independent. This means the dimension of the suspace they span is. Thus the geometric multiplicity is also.

18 The eigenvector that corresponds to the eigenvalue is. It spans a suspace of dimension one, giving geometric multiplicity of one. Thus, Property -7 holds. To see if Property -8 maintains, we need to calculate whether or not the vectors corresponding to distinct eigenvalues are orthogonal. Let v = and v =, v =. <v, v > = -() + () + () = and <v, v > = -() + () + () =. Therefore, the vectors corresponding to the distinct eigenvalues are orthogonal. Using the Gram-Schmidt process we can get the orthonormal asis = u, = u spanning the same suspace spanned y v and v. If we let = = v v u then we see that we have an orthonormal set. Property -9 applies ecause the largest eigenvalue is, and it has a multiplicity of one.

19 CHAPTER : SPECTRAL GRAPH THEORY SECTION.: INTRODUCTION Spectral graph theory is a study of the relationship etween the topological properties of a graph with the spectral (algeraic) properties of the matrices associated with the graph. The most common matrix that is studied within spectral graph theory is the adjacency matrix. L. Collatz and U. Sinogowitz first egan the exploration of this topic in 957 with their paper, Spektren Endlicher Grafen [Sp]. Originally, spectral graph theory analyzed the adjacency matrix of a graph, especially its eigenvalues. For the last 5 years, many developments have een leaning more toward the geometric aspects, such as random walks and rapidly moving Markov chains [Chu, p]. A central goal of graph theory is to deduce the main properties and the structure of a graph from its invariants. The eigenvalues are strongly connected to almost all key invariants of a graph. They hold a wealth of information aout graphs. This is what spectral graph theory concentrates on. Rememer, our definition of a graph from Chapter does not admit self-loops or multi-edges. 4

20 SECTION.: THE SPECTRUM OF A GRAPH The spectrum of a graph G is the set of eigenvalues of G, together with their algeraic multiplicities, or the numer of times that they occur. Property -: A graph with n vertices has n eigenvalues. Proof: This is a direct result of the Fundamental Theorem of Algera. The characteristic polynomial of G with n vertices is a polynomial of degree n, and the Fundamental Theorem of Algera states that every polynomial of degree n has exactly n roots, counting multiplicity, in the field of complex numers. If a graph has k distinct eigenvalues λ > λ > K > λ k with multiplicities m(λ ), m(λ ),... m(λ k ), then the spectrum of G is written Spec(G) = λ m( λ ) λ m( λ ) λn m( λn ), where k λi = n [Bi, p8] i= Example: G = a c d M = a c d a c d Figure - The characteristic polynomial is 4λ 4 4λ with eigenvalues {,,, }. Our graph has three distinct eigenvalues: -,, and, hence the spectrum of the graph G is given y Spec(G) =. 5

21 One consistent question in graph theory is: when is a graph characterized y its spectrum? [Do, p568] Properties that cannot e determined spectrally can e determined y comparing two nonisomorphic graphs with the same spectrum. Below are two graphs with the same spectrum. G = H = Figure - It is clear that the two graphs are structurally different, with different adjacency matrices, and yet they have the same spectrum, which is. We know that graphs with the same spectrum have the same numer of triangles. The example aove shows that oth graphs have no triangles, ut it also shows that graphs with the same spectrum do not necessarily have the same numer of rectangles. While the first has no rectangle, the second one has one rectangle. Our example also shows that graphs with the same spectrum do not necessarily have the same degree sequence (the sequence formed y arranging the vertex degrees in non-increasing order). The degree sequence of graph G is <4,,,, > while that of H is <,,,, >. Finally, our example also shows that connectivity cannot e determined y the spectrum. A graph is connected if for every pair of vertices u and v, there is a walk from u to v. Here the graph G is connected while graph H is not. There is a correlation etween the degree of the vertices of a graph and the largest eigenvalue, λ. When we have a k-regular graph (all vertices have degree k), then λ = k. 6

22 When we have a complete graph (all vertices are adjacent to one another) with n vertices, λ = n. If G is a connected graph, then λ is less than or equal to the largest degree of the graph. Also, λ increases with graphs that contain vertices of higher degree. In addition, the degrees of the vertices adjacent to the vertex with the largest degree affect value of λ. Example: Look at graphs G and H elow. H Figure - G Both have 5 vertices. The degree of each vertex in G is 4; in other words, G is a complete graph, so λ (G) = 4. The second graph, H, is a path; the degrees of the vertices of H are and, much smaller than that of G: λ (H) =.7. Example: Compare graphs G and H elow, and their respective λ s. In each graph, the vertex with the highest degree has degree 4, ut the adjacent vertices in G have degree, while in H they have degree. This illustrates how the degrees of the vertices adjacent to the vertex with the largest degree affect the value of λ. G λ = H λ = 7

23 SECTION.: BIPARTITE GRAPHS A ipartite graph G is one whose vertex-set V can e partitioned into two susets U and W such that each edge of G has one endpoint in U and one in W. The pair U, W is called the ipartition of G, and U and W are called the ipartition susets. Bipartite graphs are also characterized as follows: Property -: A graph is ipartite if and only if it contains no odd cycles. Proof: Let v v v k v e a cycle in a ipartite graph G; let s also assume, without loss of generality, that v U. Then, y definition of a ipartite graph, v є W, v є U, v 4 є W,... v k W, and k must e even. Therefore, the walk from v to v k is odd, and the cycle v to v is even. Let s assume G is connected and without odd cycles. The case of G disconnected would follow from this special case. Consider a fixed vertex v V(G). Let V i e the set of all vertices that are at distance i from v, that is V i = { u V(G) d(u, v ) = i}, i =,,, t}, where t is the length of the longest path in G. Clearly t is finite since G is connected, and V, V,, V t provides a partition of the vertices of G. Let s oserve that no two vertices in V are adjacent, otherwise there will e a -cycle in G. Similarly, no two vertices in V are adjacent ecause we do not have a -cycle or a 5-cycle. In fact every edge of G is of the form uv, where u V i and v V i+ for some i =,, t. Let U = V V V 4 V j, the union of all the even suscripted sets and W = V V V 5 V r+, the union of all the odd suscripted sets. This shows that G is ipartite. 8

24 Example: Graph G from Figure - elow is a ipartite graph. The vertex susets are U = {a, d}, W = {, c}. Its characteristic polynomial is λ 4-4λ and its eigenvalues are {-,,, }. G = a c d M = a c d a c d Figure - Example: Take also a look at the ipartite graph P 7, the path on 7 vertices. It has the spectrum: +,,,,,, + The eigenvalues of ipartite graphs have special properties. Figure -4 Property -: If G is ipartite graph and λ is an eigenvalue, then λ is also an eigenvalue. Proof: Let G e a ipartite graph. Let U = {u, u,... u n } and W = {w, w,... w n }, where U and W are the partite sets of V(G). Then all edges are of the form u i w j where u i U and w j W. Also, there are no edges that go from u i to u j or from w i to w j for any i, j. This makes the adjacency matrix of G A = B T B, where B is an n m matrix. B x x Because λ is an eigenvalue, we know Av = λv. So T = λ. Using simple B y y 9

25 matrix multiplication, we get By = λx. Multiplying oth sides y negative one gives us B( y) = λx. The second equation that results from the matrix multiplication is B T x = λy, and λy = ( λ)( y). Therefore, B T x = ( λ)( y). This gives us the matrix equation B T B x x = λ y y giving us the eigenvalue λ. Corollary --: The spectrum of a ipartite graph is symmetric around. [Do, p558]. This logically follows from the theorem. The eigenvalues of a ipartite graph occur in pairs of additive inverses. Recall from chapter one the general form of a characteristic polynomial is λ n + c λ n + c λ n c n Property -4: If G is a ipartite graph then c k- = for n. Proof: Since G is ipartite, y Property -, there are no odd cycles in G. Thus, c n- = for n. Let s check this with another ipartite graph. G = c a d e Figure -5 Its adjacency matrix is with characteristic polynomial λ 5 4λ + λ and eigenvalues {,,,, }. For every eigenvalue λ, λ is also an eigenvalue, as

26 stated in Theorem -, and the eigenvalues are symmetric around zero. Looking at the characteristic polynomial, c and c are oth zero as stated in Property -4.

27 SECTION.4: WALKS AND THE SPECTRUM Recall that a v i v j walk in G is a finite sequence of adjacent vertices that egins at vertex v i and ends at vertex v j. Also recall that A k i,j represents the numer of walks of length k from vertex v i to vertex v j. This means, given vertices v i and v j with dist(v i, v j ) = t in G, a graph with adjacency matrix A, we have k A i, j =, for k < t, and k A i, j. A minimal polynomial of a graph G is the monic polynomial q(x) of smallest degree, such that q(g) =. For example, if f(x) = x (x 5) (x + 4) 4, the minimal polynomial for f(x) = x(x 5)(x + 4). Property -5: The degree of the minimal polynomial is larger that the diameter. Proof: Since A is symmetric, the minimum polynomial is given y q(g:λ) = (λ - λ k ), k m, where the λ k s are all distinct. If we have m distinct eigenvalues, then the degree of the minimum polynomial is m. Suppose m d, where d is the diameter of G and suppose i and j are two vertices of G whose distance is equal to m, then the (i, j) entry of A m is positive while the (i, j) entry of A k is for all k < m. This means q(g:a), which is a contradiction, thus m > d. Example: Using the graph from Figure -, we can see that the diameter of graph G is. a G = c d

28 We found the characteristic polynomial is 4λ 4-4λ = 4λ (λ ). The minimal polynomial 4λ(λ ) has degree, which is larger than the diameter of. Property -6: If a graph G has diameter d and has m distinct eigenvalues, then m > d +. Proof: This follows from Property -5. Example: Let s look at this with Example from Chapter, elow. We have already estalished that the diameter of this graph is, ecause the distance etween any two vertices is either one or two. a Graph G Figure -6 c d The adjacency matrix for G is A =, giving us the characteristic polynomial λ 4-5λ - 4λ with the 4 eigenvalues {-.56, -,,.56}. There are 4 distinct eigenvalues, giving us m = 4. The diameter is. Therefore, m > d +. A complete graph is one in which every pair of vertices is joined y an edge. A complete graph with n vertices is denoted y K n. Example: Two isomorphic K 4 graphs are G = H= Figure -7

29 Property -7: The complete graph is the only connected graph with exactly two distinct eigenvalues. Proof: If a graph has exactly distinct eigenvalues, then its diameter has to e or. Rememer that two distinct eigenvalues means we have a minimal polynomial of degree. Also, the degree must e greater than the diameter. If the diameter is, then the graph is K. Otherwise, the graph is K p for p >. Furthermore, the degree of the characteristic polynomial of a complete graph is p (giving us p eigenvalues). Using matrix algera one can show that the characteristic polynomial for K p is ( ) p λ+ ( λ p ) + ; then is a root of the characteristic polynomial with multiplicity p and the other root, which occurs once, is p [Sc, Wednesday AM]. In the example elow, the matrix of the complete graph yields a characteristic polynomial of degree four. Note that occurs as a root times, and = 4 occurs once. Example: The adjacency matrix from Graph G in figure -5 is A =. G is a complete graph, and its characteristic polynomial is λ 4 6λ 8λ with eigenvalues {-, } where - is a triple root and is a single root. The graph has 4 vertices, 4 roots, and exactly two distinct roots. Property -8: The complete graph K p is determined y its spectrum. 4

30 Proof: This comes from Properties - and -7. If there are exactly two eigenvalues in the spectrum, then we know that the graph is a complete graph, and we know how many vertices the graph has ased on the multiplicities of the eigenvalues. Example: Given Spec G =, we know that the graph is complete with three vertices, giving us the graph G in Figure -8. G = Figure -8 5

31 CHAPTER : THE LAPLACIAN SECTION.: INTRODUCTION The Laplacian is an alternative to the adjacency matrix for descriing the adjacent vertices of a graph. The Laplacian, L, of a graph is the square matrix that corresponds to the vertices of a graph [Do, p57]. The main diagonal of the matrix represents the degree of the vertex while the other entries are as follows: A ij = - if v i and v j are adjacent otherwise The Laplacian can also e derived from D A, where D is the diagonal matrix whose entries represent the degrees of the vertices, and A is the adjacency matrix. Example: a a c d G = c d a L = c d Figure - The Laplacian of a connected graph has eigenvalues λ λ... λ n. The algeraic connectivity is defined to e λ, the second smallest eigenvalue. The name is a result of its connection to the vertex connectivity and the edge connectivity of a graph. It is the most important information contained within the spectrum of a graph [Mo, p4] and will e discussed more in Chapter 4. 6

32 The oldest result aout the Laplacian concerns the numer of spanning trees of a graph [Do, p57]. The Matrix Tree Theorem is one of the most significant applications of the Laplacian, and is usually contriuted to Kirchhoff. It is discussed in the next section. A positive semidefinite matrix is one that is Hermitian, and whose eigenvalues are all non-negative. A Hermitian matrix is one which equals its conjugate transpose. This is usually written: A H = T A = A. Example: + 4i B = 4i, then B H = B. This means B is Hermitian. Here B is Hermitian ut it is not symmetric. On the other hand, + 4i C = + 4i, is symmetric ut not Hermitian. On the other hand, a real matrix is symmetric if and only if it is Hermitian. This is ecause the every real numer is its own conjugate. Here A is the real symmetric matrix A =. If we look at each element, a ij = a ji ; its eigenvalues are {, }. The characteristic function is the function for which every suset N of X, has a value of at points of N, and at points of X N. In other words, it takes the value of for numers in the set, and for numers not in the set. Property -: The smallest eigenvalue of L is. Proof: This is a direct result of the Laplacian matrix eing a positive semidefinite matrix. It will have n real Laplace eigenvalues: = λ λ... λ n. 7

33 Property -: The multiplicity of as an eigenvalue of L is the numer of connected components in the graph. Proof: Let H e a connected component of graph G. Denote y f H R V the characteristic function of V(H). Then, L(G) f H =. Since the characteristic functions of different connected components are linearly independent, the multiplicity of the eigenvalue is at least the numer of connected components of G. Conversely, let f R V e a function from the kernel of L(G). Then f, Lf =, implying that f is constant on each connected component of G. Therefore f is a linear comination of the characteristic functions of connected components of G. [Mo, p6] Property -: The algeraic connectivity is positive if and only if the graph is connected. Proof: If λ > and λ =, then G must e connected, ecause, as mentioned aove, the eigenvalues of the Laplacian are = λ λ... λ n. This is a direct result of Property -. If G is connected, then zero is the smallest eigenvalue. λ must e greater than zero, and therefore positive. When we have a k-regular graph, a graph whose vertices all have degree k, there is a linear relationship etween the eigenvalues of the Laplacian and the eigenvalues of the adjacency matrix A. If θ θ... θ n are the eigenvalues of L (Laplacian eigenvalues of G) and λ, λ,... λ n are the eigenvalues of A (the adjacency matrix of G), then θ i = k λ i. This is a result of the graph eing a regular graph, giving us the relationship L = ki A etween the Laplacian and the adjacency matrix. No such relationship exists etween the eigenvalues of the Laplacian and adjacency matrix of a non-regular graph. 8

34 SECTION.: SPANING TREES In order to discuss spanning trees, we must first cover a few definitions. A tree is a connected graph that has no cycles. Example: Figure - A tree Not a tree A sugraph H of a graph G is a graph whose vertex and edge sets are suset of V(G) and E(G) in that order. A few sugraphs of the tree aove are: Figure - A sugraph H is said to span a graph G if V(H) = V(G). A spanning tree of a graph is a spanning sugraph that is a tree. Given graph G elow, graph H is a spanning tree of G. 9

35 Figure -4 Graph G Graph H Before we go into the next few properties, we need to understand a cofactor of a matrix, which egins with the minor of a matrix. A minor M ij of a matrix B is determined y removing row i and column j from B, and then calculating the determinant of the resulting matrix. The cofactor of a matrix is ( ) i + j M ij. Example: B = Minor M comes from deleting row and column, and then calculating the determinant of the resulting matrix, M = Cofactor C = ( ) + M = Also, given an n n matrix B, det(b) = a j C j + a j C j a nj C nj = a i C i + a i C i a in C in = the sum of the cofactors multiplied y the entries that generated them. Property -4, The Matrix Tree Theorem: Given a graph G, its adjacency matrix A, and its degree matrix C, the numer of nonidentical spanning trees of G is equal to the value of any cofactor of the matrix C A.

36 Proof: Let D = C A. The sum of the elements of any row i or any column i of A is the degree of vertex v i. Therefore, the sum of any row i or column i of matrix D is zero. It is a result of matrix theory that all cofactors of D have the same value. Let G e a disconnected matrix of order p. Let G n e a sugraph of G whose vertex set is {v, v,..., v n }. Let D e the (p ) (p ) matrix that results from deleting row p and column p from matrix D. Since the sum of the first n rows of D is the zero vector with p entries, the rows of D are linearly dependent, implying that det(d ) =. Hence, one cofactor of D has value zero. Zero is also the numer of spanning trees of G, since G is disconnected. Now let G e a connected (p, q) graph where q p. Let B denote the incidence matrix of G and in each column of B, replace one of the two nonzero entries y. Denote the resulting matrix y M = [m ij ]. We now show that the product of M and its transpose M T is D. The (i, j) entry of MM T is q k = m ik m jk, which has the value deg v i if i = j, the value if v i v j є E(G), and otherwise. Therefore, MM T = D. Consider a spanning sugraph H of G containing p edges. Let M e the (p ) (p ) sumatrix of M determined y the columns associated with the edges of H and y all rows of M with one exception, say row k. We now need to determine the determinant of M. If H is not connected, then H has a component, H*, not containing v k. The sum of the row vectors of M corresponding to the vertices of H* is the zero vector with p entries. Hence, det M =. Now assume H is connected. H is then a spanning tree of G. Let h v k e an end-vertex of H, and e the edge incident with it. Also, let h v k e an end-vertex of the tree

37 H h and e the edge of H e incident with e. We continue this process until finally '' only v k remains. A matrix M = [ m ] can now e otained y a permutation of the rows ij '' and columns of M such that m = if and only if u i and e i are incident. From the ij manner in which M was defined, any vertex u i is incident only with edges e j, where j i. '' This, however, implies that M is lower triangular, and since m = for all i, we conclude that det(m ) =. However, the permutation of rows and columns of a matrix affects only the sign of its determinant, implying that det(m ) = det(m ) =. Since every cofactor of D has the same value, we evaluate only the i th principal cofactor. That is, the determinant of the matrix otained y deleting from D row i so that the aforementioned cofactor equals det(m i M T i ), which implies that this numer is the sum of the products of the corresponding major determinants of M i and M T i. However, corresponding major determinants have the same value and their product is if the defining columns correspond to a spanning tree of G and is otherwise. [Cha, p66] A key result to the Matrix Tree Theorem is Cayley s Tree Formula. Corollary -4-, Cayley s Tree Formula: The numer of different trees on n laeled vertices is n n. Proof: Start with the matrix (D A) of a complete graph; we get an n n matrix M ij n, i = j with m ij =, otherwise. Calculating the determinant of a cofactor of M gives us n n. From the Matrix Tree Theorem this numer is the numer of spanning trees.

38 SECTION.: AN ALTERNATIVE APPROACH There is an alternative way of defining eigenvalues. We can define them in their normalized form. One advantage to this definition is that it is consistent with eigenvalues in spectral geometry and in stochastic processes [Chu, p]. It also allows results which were only known for regular graphs to e generalized to all graphs. We will use NL to represent the Laplacian calculated using this definition. In a graph where d v represents the degree of vertex v, the Laplacian would e defined to e the matrix NL(u, v) if u = v and d v if u and v and adjacent d u d v otherwise Example: Graph G a c d 6 NL = Figure -5 When graph G is a k-regular graph, it is easy to see that NL = I k A. The degree of each vertex is k, so d u d v = k. Every entry of A that has a will ecome k A, and when we sutract that from I, we get k for all vertices that are adjacent. The non-adjacent

39 elements will e for oth I and A. The diagonal entries of I are, and the diagonals of A (and therefore A) are, giving us when we sutract I A. k k Example: Graph G a c d NL = = I A Figure -6 Let S e a matrix whose rows represent the vertices of a graph G and whose columns represent the edges of G. Each column corresponding to an edge e = uv has an entry of in the row corresponding to vertex u and an entry of d u dv in the row corresponding to vertex v. All other entries in the column will e zero. Property -5: NL = S S T. Proof: This is a direct result of their definitions. When we multiply an element u of S y an element v of S T, and there exists an edge uv, then we are multiplying d u d v = d u d v = corresponding element in NL. If there is no edge uv, then the entries in S, S T, and NL are zero. Example: a G = Figure -7 c d 4

40 5 NL = S = d c a S S T = = = NL Property -6: is an eigenvalue of NL. Proof: Let g denote an aritrary function which assigns to each vertex v of G a real value g(v). We can view g as a column vector. Then = g g g LT T g g g Lg g,,,, / / where T = the diagonal matrix where the entries are the degrees of the vertices, and L is the regular Laplacian. g = T / f and = u~v the sum over all unordered pairs {u, v} for which u and v are adjacent. x x g x f ) ( ) ( denotes the standard inner product in n R. Then = g g g LT T g g g Lg g,,,, / / = f T f T Lf f / /,, = v v v u d v f v f u f ~ ) ( )) ( ) ( ( [Chu, p 4], which = when f(u) = f(v). This also shows us how all eigenvalues are nonnegative. Therefore the spectrum of NL is the set of eigenvalues = λ λ λ n-. a ac c cd

41 When we use this normalized definition of the Laplacian for a graph with n vertices and no isolated vertices, a few properties unfold. Property -7: λ Proof: i i i i n λ = tr(nl), the trace of NL. If the graph is connected, then tr(nl) = n, ecause each element of the diagonal is, and we have an n n matrix. If the graph is disconnected, then one or more elements of the diagonal will e zero, causing the sum of the diagonals to e less than n, and giving us a tr(nl) < n. Property -8: For n, λ n n and λ n- n n Proof: Assume that λ > n ; since all the remaining n - eigenvalue in the spectrum n are at least as large as λ their sum λ n n. Similarly assume that λ n- < i λ > n. This contradicts Property.7; thus i n ; since this eigenvalue is the largest in the n spectrum, and λ is estalished to e, the sum of the n eigenvalues i λ < n. This again i contradicts Property.7 and therefore λ n- n. n Let s look at Figure -8, a very simple example, and apply these properties. G = a c NL = Figure -8 6

42 The characteristic polynomial is λ - λ + λ, yielding eigenvalues {,, }. Property -6 holds, since is an eigenvalue. The sum of the eigenvalues is, so property -7 holds, and λ =, which demonstrates Property -8. 7

43 CHAPTER 4: APPLICATIONS SECTION 4.: CHEMICAL APPLICATIONS Recall that the eigenvalues of L(G), the Laplacian matrix of graph G are λ λ... λ n where λ n = and λ n > if and only if G is connected. Also recall that a tree is a connected acyclic graph. A chemical tree is a tree where no vertex has a degree higher than 4. Chemical trees are molecular graphs representing constitutional isomers of alkanes. If there are n vertices, each chemical tree represents a particular isomer of C n H n+. The first four are methane, ethane, propane, and utane. After that, the alkanes are named ased on Greek numers. For example, C 5 H is pentane. Compounds whose carons are all linked in a row, like the two elow, are called straight-chain alkanes. For example, if n =, we have the graph in Figure 4-, which represents methane. = Caron = Hydrogen Figure 4- Methane Figure 4- shows us utane, which is C 4 H. Figure 4- Butane 8

44 Compounds that have the same formula, ut different structures, are called isomers. When C 4 H is restructured as in Figure 4-, we have isoutane, or - Methylpropane. Butane and -Methylpropane are isomers. Figure 4- Isoutane or -Methylpropane Compounds with four carons have isomers, while those with five carons have isomers. The growth, however, is not linear. The chart elow compares the numer of carons with the numer of isomers. Formula Numer of Isomers C 6 H 4 5 C 7 H 6 9 C 8 H 8 8 C 9 H 5 C H 75 C 5 H 4,47 C H 4 66,9 [Mc, p76] When a caron has four carons onded to it, we have a quarternary caron. An example is elow in Figure 4-4, which is called a,-dimetylpropane. It is isomeric to Pentane. 9

45 Figure 4-4 For simplicities sake, we will just draw the caron atoms from this point on, with the understanding that there are enough hydrogen atoms attached to each caron to give that caron atom a degree of 4. Study was done on the eigenvalues of molecular graphs, and in particular, λ, the largest eigenvalue of a graph. When the isomeric alkanes are ordered according to their λ values, regularity is oserved. [Gu, p 48] Let Δ denote the maximum degree of a graph. The chemical trees that pertain to the 8 isomeric octanes C 8 H 8 follow a pattern with respect to their largest eigenvalue, λ. The isomer with the smallest λ (.8478) value is the straight-chain octane in Figure 4-5, that has Δ =. Figure 4-5 The next isomers have various extensions of ranching, ut none possess a quaternary caron atom. All of them have Δ =, and their λ s are greater than that of the straight-chain graph in Figure 4-5, where Δ =, and less than the following seven, who have Δ = 4. They are shown elow in Figure

46 Figure 4-6 The th through the 8 th octanes contain a quaternary caron atom, they all have Δ = 4, and they have the largest λ. The largest one has λ = and is the last tree shown elow in Figure 4-7. Figure 4-7 This same regularity occurs with isomeric alkanes with n caron atoms, discussed aove. The normal alkane with Δ = has the smallest λ. All alkanes with Δ = have λ greater than the alkanes with Δ =, and smaller than any isomer with Δ = 4. We can therefore draw the conclusion that Δ, which tells us whether or not there is a quaternary 4

47 caron atom, is the main molecular structure descriptor affecting the value λ, the largest Laplacian eigenvalue of an alkane. It has een discovered that λ can e ounded y Δ + < λ < Δ + + Δ Also, y using a linear comination of the lower and upper ounds, λ can e estimated y λ Δ + + γ Δ, where γ depends on oth n and Δ. For alkanes, it has een discovered through numerical testing that γ.. [Gu p 4] It is possile to estalish the alkane isomers with Δ = or Δ = 4 that have the minimal λ. Give P n, elow, T n min is the tree that estalishes the minimal λ for Δ =, and Q n min is the tree that estalishes the minimal λ for Δ = 4. P n = n... T min n =... n Q min n n =... Figure 4-8 max The structure trees that represent the maximal λ are more complex. The T n and Q max n coincide with the chemical trees that have the same Δ and n, having maximal λ and minimal W, where W represents the Wiener topological index of alkanes, and n conforms to the formula W = n i= λ complex, and will not e covered here. i. The exact characterizations of these trees are 4

48 SECTION 4.: PROTEIN STRUCTURES The three dimensional structure of proteins is the key to understanding their function and evolution [Vi, p]. There are an enormous numer of proteins in nature, ut a limited numer of three-dimensional structures that represent them. A protein is formed y the sequential joining of amino acids, end-to-end, to form a long chain-like molecule, or polymer. These polymers are referred to as polypeptides. There are four major protein classes, shown elow in Figure 4-9. [Vi, p] The cylinders represent helices (plural for helix) and the arrows represent strands. A helix is a spiral molecule formed from enzene rings. Two of the current challenges are identifying the fold adopted y the polypeptide chain, and identifying similarities in protein structures. Figure 4-9 4

49 Analysis of stale folded three-dimensional structures provides insights into protein structures for amino acid sequences and drug design studies. The geometry of a protein structure is the composition of the protein ackone and side-chains. Protein structures can have the same gross shape, ut have different geometries. Graphs have helped represent the topology of protein structures, no matter how complex. The main prolem is to define the vertices and edges. Below are a few simple examples of proteins with their graphs elow them. Β-Hairpin Greek key βαβ Figure 4- Properties of graphs and their graph spectral give information aout protein structure, depending on how the vertices and edges are defined. The asic unit of a protein is its amino acid residue. To study cluster identification, the amino acids represent the vertices and the three-dimensional connectivity etween them is represented y the edges. To study fold and pattern identification and the folding rules of proteins, α-helixes and β-strands are used for vertices and spatially closed structures are used for edges. To identify proteins with similar folds, the ackones are the vertices and the spatial neighors within a certain radius are the edges. 44

50 Mathematical graphs are used to represent β structures, which is much more advantageous that drawing three-dimensional figures. The vertices represent the single β-strands and the two edge sets represent the sequential and hydrogen ond connections. Connected graphs are used to represent α-helical structures. The vertices represent secondary structures and the edges represent contacts etween helices. The main reason for setting the structures up this way was to gain information aout the folding process of protein structures and understand etter the organization and patterns within these structures. It is also used to compare protein structures. The protein connectivity is determined y identifying the main chain atoms, which is found due to the closeness within a prescried distance. This comes from identifying clusters. Two protein graphs can e compared to see if they have common features, and thus provide insight into structural overlaps of proteins. One way is a tree searching algorithm, which is a series of matrix permutations. Through it, sugraph isomorphisms are detected to determine the largest sugraph that is common in a pair of graphs. This can highlight areas of structural overlap & therefore show structural and functional commonalities not found through other methods. Unfortunately, this method requires a very high numer of computations, ut a few heuristic methods have een discovered to reduce the time and cost of the computations. Structural iologists are continuously finding promising applications of graph theory with optimism that this field of mathematics will continually contriute sustantially to the understanding of protein structure, folding staility, function and dynamics. Certainly, there is more to come. 45

51 SECTION 4.: GRAPH COLORING One of the classic prolems in graph theory is vertex-coloring, which is the assignment of colors to the vertices of a graph in such a way that no two adjacent vertices have the same color. The oject is to use as few colors as possile. The proper coloring of a graph also forms a natural partition of the vertex set of a graph. The chromatic numer, χ(g), is the least numer of colors required for such a partition. A graph G is l-critical if χ (G) = l and for all induced sugraphs Λ G we have χ (Λ) < l. The spectrum of a graph gives us insight into the chromatic numer. Before we can understand the property that provides an upper-ound for χ(g), we must first estalish Property 4-. Property 4-: Given a graph G with χ (G) = l, there exists a sugraph of G, Λ G, such that χ (Λ) = l, and every vertex of Λ has degree l in Λ. Proof: The set of all induced sugraphs of G is non-empty and contains some graph whose chromatic numer is l (including G itself). The set also contains some sugraphs whose chromatic numer is not l. An example would e a graph with one vertex. Let Λ e a sugraph such that χ (Λ) = l, and χ (G) is minimal with respect to the numer of vertices. Then Λ is l-critical. If v is any vertex of Λ, then V ( Λ) \ v is an induced sugraph of Λ and has a vertex-coloring with l colors. If the degree of v in Λ were less than l, then we could extend this vertex coloring to Λ, contradicting the fact that χ (Λ) = l. Thus the degree of v is at least l. Property 4-: For any graph G, χ (G) + λ, where λ is the largest eigenvalue of the adjacency matrix of G. 46

52 Proof: From Property 4-, there is an induced sugraph Λ of G such that χ (G) = χ (Λ) and d min (Λ) χ (G), where d min (Λ) is the least degree of the vertices of Λ. Thus we have χ (G) + d min (Λ) + λ (Λ) + λ (G). [Bi, p55] Of course, the asolute largest value of χ (G) is n, the numer of vertices. If λ < n, then we will have a smaller maximum than n. Property 4-: The lower ound of the chromatic numer is χ (G) + λ λ min. Proof: The vertex set V(G) can e partitioned into χ (G) coloring classes. Thus, the adjacency matrix A of G can e partitioned into χ sumatrices. This was seen in Chapter. In this case, the diagonal sumatrices A ii ( i v) consists entirely of zeros, and so λ (A ii ) =.. It is known that λ (A) + (t )λ min (A) t i= λ max ( A ii ). Therefore, we have λ(a) + (χ(g) )λ min (A). But, if G has at least one edge, then λ min (A) = λ (G) <. [Bi, p57] A classical application of vertex coloring is the coloring of a map. To make the process as inexpensive as possile, we want to use a few colors as possile. In the graph, the countries are represented y vertices with edges drawn etween those countries that are adjacent. Determining the chromatic numer of the graph gives us the fewest colors necessary to color the map. Another application to graph coloring is a sorting prolem, such as sorting fish in a pet store. Some fish can e in the same tank together, while other fish cannot. Say we have fish types A, B, C, D, E and F. They can e put into tanks according to the chart elow 47