Cospectral bipartite graphs for the normalized Laplacian. Steven Paul Osborne. A dissertation submitted to the graduate faculty

Transcription

1 Cospectral bipartite graphs for the normalized Laplacian by Steven Paul Osborne A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Steven Butler, Co-major Professor Leslie Hogben, Co-major Professor Elgin Johnston Ryan Martin Yiu Tung Poon Iowa State University Ames, Iowa 2013

2 ii DEDICATION I would like to dedicate this thesis to my wife Lacey and to my daughter Eva. Their support made this research possible.

3 iii TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES vi CHAPTER 1. General Introduction Definitions and Notation Spectral Graph Theory Adjacency Matrix Laplacian Matrix Signless Laplacian Matrix Normalized Laplacian Matrix Dissertation Organization CHAPTER 2. Constructions of cospectral bipartite graphs for the normalized Laplacian Introduction Eigenvectors of the adjacency matrix of trees An example of an infinite family of cospectral pairs of trees Using the weighted normalized Laplacian Characterization of diameter 3 trees Balanced two-weighted paths An example of cospectral bipartite graphs Concluding Remarks CHAPTER 3. Almost all trees are normalized Laplacian cospectral Introduction

4 iv 3.2 Characteristic polynomial Almost all trees are cospectral CHAPTER 4. General Conclusions General Discussion Recommendations for Future Research ACKNOWLEDGEMENTS

5 v LIST OF TABLES 2.1 Number of trees on n vertices, f n, number of trees on n vertices cospectral with another tree with respect to the normalized Laplacian, g n, number of trees on n vertices cospectral with another tree with respect to the adjacency matrix, h n The number of trees cospectral with another tree with respect to the adjacency, (signless) Laplacian, and normalized Laplacian matrices... 42

6 vi LIST OF FIGURES 1.1 Cospectral graphs Cospectral connected graphs Graphs cospectral with respect to Laplacian matrix Graphs cospectral with respect to signless Laplacian matrix Graph cospectral with a proper subgraph with respect to normalized Laplacian matrix T 1 = {8, 11} and T 2 = {1, 2, 5, 7, 8, 12} are every-other trees. The leaves of T 2 are 1, 2, 8, P W G G G H H H T T T T T T

7 vii 2.16 A cospectral family D 3,3, D 3,4, P 3,3, P 3,4, (S 1, v 1 ) (S 2, v 2 ) (T 1, v 1 ) (T 2, v 2 ) (G 1, v 1 ) (G 2, v 2 ) (H 1, v 1 ) (H 2, v 2 )

8 1 CHAPTER 1. General Introduction Spectral graph theory attempts to answer a simple question: If we have a large data structure, what information can we capture with relatively few numbers? Specifically, can we go from storing the graph (n 2 data points) to the spectrum (n data points) of a matrix associated with the graph, or even just a few eigenvalues of large and small magnitude? An immediate question arises: Can the eigenvalues of the adjacency matrix 1 of a graph be used to completely reconstruct the graph (the Inverse Eigenvalue Problem )? It is easy to show that this is not possible. The graphs in Figure 1.1 have the same spectrum ( 2, 0, 0, 0, 2). One of these graphs is disconnected, so it might seem that this is an anomaly associated with the connectedness of a graph. However, there are two connected graphs on six vertices which are cospectral (Figure 1.2). Now, one might be tempted to think that the spectrum is a poor gauge of a graph. But there are many graphs which can be recovered via their spectrum. In fact, there are papers by van Dam and Haemers which address this subject [9, 10]. Figure 1.1 Cospectral graphs Spectral graph theory has a rich background. See [2] and [8] for an overview of spectral graph theory and [3] and [5] for an overview as concerned with the normalized Laplacian. We say a graph G is determined by its spectrum (DS) for a particular type of matrix associated with the 1 Formal definitions of a graph and the matrices associated with it are given in Section 1.1

9 2 Figure 1.2 Cospectral connected graphs graph if there is no nonisomorphic graph which has the same spectrum for that type of matrix. Conversely, we say two graphs G and H are cospectral with respect to a given type of matrix associated with the graph if these matrices share the same spectrum including multiplicity. It is known that many graphs have a cospectral mate [13, 16]. Clearly, nonisomorphic graphs with the same spectrum demonstrate the weaknesses of a certain type of matrix. In order to understand these weaknesses it is helpful to look at the structure of cospectral graphs. For the normalized Laplacian, little is known about cospectral pairs; see [4] for a few constructions of cospectral graphs for the normalized Laplacian. 1.1 Definitions and Notation A graph G = (V (G), E(G)) is a pair of sets of vertices V (G) and edges E(G). If G is understood, we often write V = V (G) and E = E(G). An edge is a two element subset of V. If the pair {u, v} E, we say that u and v are adjacent in G or they are neighbors in G. The neighborhood of u in G, denoted N G (u), is the set of neighbors of u in G. The degree of a vertex v in G, denoted deg G v, is the cardinality of its neighborhood in G. An isolated vertex v in G is a vertex with deg G v = 0. A subgraph of G is a graph (V, E ) where V V and E E. The path of order n, P n, is the graph with V = {v 1, v 2,..., v n } and E = {{v 1, v 2 }, {v 2, v 3 },..., {v n 1, v n }}. A graph is connected if for all vertices u and v in G there is a path in G beginning at u and ending at v. If a graph is not connected, we say it is disconnected. A component of G is a maximal connected subgraph of G. The cycle of length n, C n, is the graph with V = {v 1, v 2,..., v n } and E = {{v 1, v 2 }, {v 2, v 3 },..., {v n 1, v n }, {v n, v 1 }}.

10 3 A connected graph is a tree if it does not contain a cycle as a subgraph. A graph is bipartite if V = X Y and N G (x) Y for all x X and N G (y) X for all y Y for some partition X and Y of V. A well known result of graph theory is that a graph is bipartite if and only if it contains no cycles of odd length. Thus, a tree is necessarily bipartite. A bipartite component of G is a component of G that is bipartite. Given a matrix N, the characteristic polynomial of N is p N (x) := det(xi N). The spectrum of a matrix N is the multiset of roots of p N (x) including multiplicity. The diagonal matrix denoted by diag(a 1, a 2,..., a n ) is the matrix with entries a i if i = j, and diag(a 1, a 2,..., a n ) ij = 0 otherwise. Although there are many different ways of interpreting a graph as a matrix, we are particularly concerned with four variants: adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices. The adjacency matrix of a graph G, denoted A(G), is the symmetric matrix indexed by the ordered set (v 1,..., v n ) of vertices of G with (A) uv = 1 if u and v are adjacent in G and 0 otherwise. The diagonal degree matrix of G is D(G) := diag(deg G v 1,..., deg G v n ). The Laplacian matrix of G is L(G) := D(G) A(G), the signless Laplacian matrix of G is Q(G) := D(G) + A(G), and the normalized Laplacian matrix of G, L(G) is given by 1 if u = v and deg v 0, L(G) uv = 1 deg u deg v if u v, and 0 otherwise. Thus, if G has no isolated vertices L(G) = D(G) 1/2 L(G)D(G) 1/2. Note that A(G) is often written A if the graph G is implied, and this applies to D(G), L(G), Q(G) and L(G) as well. Define M(λ, t, G) := λi n A(G) + td(g) where G is a graph on n vertices. The generalized characteristic polynomial of G is φ(λ, t, G) := det(m(λ, t, G)). Note that if G has no isolated vertices, φ(x, 0, G) = p A(G) (x), φ( x, 1, G) = ( 1) n p L(G) (x), φ(x, 1, G) = p Q(G) (x), and φ(0, x + 1, G) = ( 1) n det(d)p L(G) (x). Thus if two graphs have the same generalized characteristic polynomial, they are simultaneously cospectral with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices.

11 4 1.2 Spectral Graph Theory Each of these types of matrices comes with a set of strengths and weaknesses. In the table below we summarize the four types of matrices and four different structural properties of a graph and indicate which properties can be determined by the eigenvalues. A No answer indicates the existence of two non-isomorphic graphs which have the same spectrum but differ in the indicated structure. This table may also be found in [3]. Matrix Bipartite # Components # Bipartite Components # Edges Adjacency Yes No No Yes Laplacian No Yes No Yes Signless Laplacian No No Yes Yes Normalized Laplacian Yes Yes Yes No We will now discuss this table in detail Adjacency Matrix As stated in the table, the adjacency matrix can detect whether a graph is bipartite. In fact, it just comes down to inspecting the symmetry of the spectrum. Theorem [6] If λ 1... λ n are the eigenvalues of A(G), then G is bipartite if and only if λ i = λ n+1 i for i = 1,..., n. While the adjacency can tell if a graph is bipartite, it is incapable of determining the number of bipartite components in a graph. The Saltire pair is a demonstration of this fact. Note that this pair also implies that the adjacency matrix cannot count the number of components. However, it can count edges. Theorem [6] If λ 1... λ n are the eigenvalues of the adjacency matrix of a graph with n vertices and m edges, then i λ2 i = 2m Laplacian Matrix The graphs shown in Figure 1.3 are cospectral with respect to the Laplacian matrix. They demonstrate that the Laplacian matrix cannot detect whether a graph is bipartite. This also

12 5 implies the Laplacian matrix is incapable of counting the number of bipartite components of a graph. However, the Laplacian matrix can count the components and edges of a graph. Figure 1.3 Graphs cospectral with respect to Laplacian matrix Theorem [5] If the multiplicity of 0 as an eigenvalue of L(G) is c, then G has c components. The following is a well known result. Theorem If λ 1... λ n are the eigenvalues of the Laplacian matrix of a graph n vertices and m edges, then i λ i = 2m. Proof. We know tr(l) = tr(d) = deg v = 2m. However, tr(l) = i λ i. v V (G) Signless Laplacian Matrix The signless Laplacian can detect neither if a graph is bipartite nor the number of its components. The graphs in Figure 1.4 are cospectral with respect to the signless Laplacian matrix and demonstrate this fact. However, the signless Laplacian is capable of counting the number of bipartite components in a graph. Theorem [7] If the multiplicity of 0 as an eigenvalue of Q(G) is b, then G has b bipartite components. The following is a well known result.

13 6 Figure 1.4 Graphs cospectral with respect to signless Laplacian matrix Theorem If λ 1... λ n are the eigenvalues of the signless Laplacian matrix of a graph with n vertices and m edges, then i λ i = 2m. The proof is identical to the proof of Theorem Normalized Laplacian Matrix On our short list of attributes, the normalized Laplacian detects three of the four characteristics. The follow theorems are due to Chung [5]. Theorem [5] Assume G is a graph and λ 1... λ n are the eigenvalues of L(G). The following statements are equivalent: 1. G is bipartite. 2. G has i components and λ j = 2 for 1 j i. 3. For each λ i, 2 λ i = λ j for some j. Theorem [5] If the multiplicity of 0 as an eigenvalue of L(G) is c, then G has c components. In order to show that the normalized Laplacian can count the number of bipartite components of a graph, we make the following simple observation: the normalized Laplacian spectrum of a graph is the union of the normalized Laplacian spectra of its components [5]. Theorem If λ 1... λ n are the eigenvalues of L(G) where α is the multiplicity of 2, then the number of bipartite components of G is α + n i λ i.

14 7 Proof. We wish to count the number of bipartite components of a graph G using the spectrum of its normalized Laplacian matrix. The cardinality of the spectrum is n. This tells us G has n vertices. Further, we know that the diagonal entry in L(G) corresponding to each vertex in G is 1 unless the vertex is isolated (then the value is 0). Thus, the number of isolated vertices in G is l = n tr(l) = n i λ i. Thus, we may remove l zeros from the spectrum and our spectrum now corresponds to G with all isolated vertices removed. By Theorem 1.2.8, we know each component must have a exactly one 0 in its spectrum. Thus, by Theorem 1.2.7, each bipartite component must have exactly one 2 in its spectrum. Thus, α + l is the number of bipartite components in G. Unfortunately, the normalized Laplacian is not capable of counting the edges of a graph. The example given in Figure 1.5 shows a graph cospectral with one of its proper subgraphs with respect to the normalized Laplacian. Figure 1.5 Graph cospectral with a proper subgraph with respect to normalized Laplacian matrix 1.3 Dissertation Organization This dissertation is organized in the format of a dissertation containing journal papers. In the general introduction, pertinent background information is presented. Chapter 2 contains the paper Constructions of cospectral bipartite graphs for the normalized Laplacian [14] submitted to the Electronic Journal of Linear Algebra. In the paper, we

15 8 construct two infinite families of trees that are pairwise cospectral with respect to the normalized Laplacian. We also use the normalized Laplacian applied to weighed graphs to give new constructions of cospectral pairs of bipartite unweighted graphs. Chapter 3 contains the paper Almost all trees are normalized Laplacian cospectral [15] submitted to Linear Algebra and its Applications. In the paper, we show that almost all trees have a cospectral mate for the normalized Laplacian matrix as well. We also show that almost every tree is cospectral with another tree with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices simultaneously by showing that almost all trees have a mate with the same generalized characteristic polynomial. Chapter 4 is for general conclusions. Results are summarized and recommendations for future research are also presented.

16 9 Bibliography [1] N. Biggs. Algebraic Graph Theory. Cambridge University Press, Cambridge, UK, [2] A. Brouwer and W. Haemers. Spectra of Graphs. Springer, New York, [3] S. Butler and F. Chung. Spectral Graph Theory, in Handbook of Linear Algebra, 2nd Edition. L. Hogben, Editor. CRC Press, Boca Raton, FL, [4] S. Butler and J. Grout. A construction of cospectral graphs for the normalized Laplacian. Electronic Journal of Combinatorics, 18:#231, 20pp, [5] F. Chung. Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92, AMS, Providence, RI, [6] D.M. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs: Theory and Applications. Academic Press, New York, (3rd Edition, Johann Abrosius Barth Verlag, Heidelberg- Leipzig, 1995.) [7] D. Cvetković, P. Rowlinson, S. Simić. Signless Laplacians of finite graphs. Linear Algebra and its Applications, 423(1): , [8] D. Cvetković, P. Rowlinson, S. Simić. An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge, UK, [9] E.R. van Dam and W. Haemers. Which graphs are determined by their spectrum? Linear Algebra and its Applications, 373: , [10] E.R. van Dam and W. Haemers. Developments on spectral characterizations of graphs. Discrete Mathematics, 309: , 2009.

17 10 [11] J-M. Guo, J. Li, W.C. Shiu. On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph. Czechoslovak Mathematical Journal, to appear. [12] F. Li, H. Lu, W. Wang, Z. Xu. Graphs determined by their generalized characteristic polynomials. Linear Algebra and its Applications, 434: , [13] B. D. McKay. On the spectral characterisation of trees. Ars Combinatoria, 3: , [14] S. Osborne. Constructions of cospectral bipartite graphs for the normalized Laplacian. Under review. [15] S. Osborne. Almost all trees are normalized Laplacian cospectral. Under review. [16] A. Schwenk. Almost all trees are cospectral. New directions in the theory of graphs, , [17] W. A. Stein, et al. Sage Mathematics Software (Version 5.12). The Sage Development Team,

18 11 CHAPTER 2. Constructions of cospectral bipartite graphs for the normalized Laplacian A paper submitted to the Electronic Journal of Linear Algebra Steven Osborne Abstract We construct two infinite families of trees that are pairwise cospectral with respect to the normalized Laplacian. We also use the normalized Laplacian applied to weighed graphs to give new constructions of cospectral pairs of bipartite unweighted graphs. 2.1 Introduction The spectrum of graphs is a well studied problem [1, 4, 5]. We say two graphs G and H are cospectral with respect to a given matrix described by the graph (e.g., adjacency, Laplacian, etc.) if these matrices share the same spectrum including multiplicity. It is known that many graphs have a cospectral mate [8, 9]. However, it is difficult to determine which graphs do not have a cospectral mate. The goal of spectral graph theory is to know how much information about the graph we can determine given its spectrum [7]. Clearly, distinct graphs with the same spectrum demonstrate the weaknesses of a certain matrix. In order to understand these weaknesses it is helpful to look at the structure of cospectral graphs. For the normalized Laplacian little is known about cospectral pairs; see [3] for constructions of cospectral graphs for the normalized Laplacian. In this paper we will give some new constructions of bipartite

19 12 graphs which are cospectral for the normalized Laplacian, including the first known example of an infinite family of cospectral trees. We will consider the spectrum of a graph as it pertains to the normalized Laplacian matrix. We will first consider simple graphs and then we will introduce weighted graphs in Section 2.4. The adjacency matrix of a graph G, denoted A(G), is the matrix indexed by the ordered set (v 1,..., v n ) of vertices of G with (A) uv = 1 if u and v are adjacent in G and 0 otherwise. The diagonal degree matrix of G is D(G) := diag(deg G v 1,..., deg G v n ). The Laplacian matrix of G is L(G) := D(G) A(G) and the normalized Laplacian matrix of G is L(G) := D(G) 1/2 L(G)D(G) 1/2 where by convention each isolated vertex contributes a 0 to the spectrum of L(G). Thus, L(G) uv = 1 if u = v and deg v 0, 1 deg u deg v if u v, and 0 otherwise. See [4] for an overview of the normalized Laplacian. Note that A(G) is often written A if the graph G is implied, and this applies to D(G), L(G) and L(G) as well. The distance between two vertices u and v in G, denoted d G (u, v), is the length of the shortest path connecting u and v in G. The diameter of a graph G is diam(g) := max u,v G d G (u, v). The neighborhood of a vertex v in G, denoted N G (v), is the set of vertices adjacent to v in G. Two vertices u and v are twins in a graph G if N G (v) = N G (u) (this implies u and v are not adjacent) Eigenvectors of the adjacency matrix of trees We know that 0 and 2 appear in the spectrum of the normalized Laplacian of a tree [4]. However, 1 often appears with relatively high multiplicity. As L = I D 1/2 AD 1/2, a 0 in the spectrum of A implies a 1 in the spectrum of L and a set of linearly independent eigenvectors for eigenvalue 0 of A are related to a set of linearly independent eigenvectors for eigenvalue 1 of L. Therefore, we begin by examining the structure of 0-eigenvectors of the adjacency matrix of a tree. Definition An every-other tree, denoted T eo, is a set of vertices of a tree of T such that

20 Figure 2.1 T 1 = {8, 11} and T 2 = {1, 2, 5, 7, 8, 12} are every-other trees. The leaves of T 2 are 1, 2, 8, for every v T, N T (v) T eo {0, 2}, and 2. for every vertex u T eo, there exists at least one vertex w T eo such that d T (u, w) = 2. Define a leaf of T eo to be a vertex u T eo such that d T (u, w) > 2 for all but one w T eo. The vector x with ±1 for each v in T eo such that vertices in T eo that are distance two apart in T have opposite sign and 0 elsewhere is the standard representation vector for T eo. See Figure 2.1 for examples. Note that it is a simple observation that any every-other tree must have at least two leaves. Also, the standard representation vector x for T eo can be constructed with no conflicting signs by assigning a leaf of T eo the value 1 in x. Given the structure of T eo, the rest of x is now completely determined. We will eventually be able to decompose 0-eigenvectors of A(T ) using every-other trees. For now, we will investigate the properties of an every-other tree. Proposition Given a tree T, an every-other tree T eo of T, and v T eo, then 1. v is not adjacent in T to any vertex in T eo, and 2. v is a leaf of T eo if and only if v is a leaf of T. Proof.

21 14 1. Assume not. Then as v must not be adjacent to only one vertex in T eo there exists two vertices u 1 and u 2 in T eo such that v is adjacent to u 1 and u 2. Now, u 1 and u 2 are not adjacent and N T (u 2 ) T eo = 2, so there exists a u 3 in T eo such that u 2 and u 3 are adjacent. Continuing in this fashion we can create an infinite sequence of unique vertices, which is a contradiction. 2. Assume v is a leaf of T eo and that u T eo such that d T (v, u) = 2. Assume that v, w, u is the path of length 2 in T. Now, if v were to have a neighbor w w in T, then w must have a neighbor u v, u in T such that u T eo, thus contradicting the fact that v is a leaf of T eo. Hence, v must be a leaf of T. Now, assume that v is not a leaf of T eo but that v is a leaf of T. Let w be the neighbor of v in T. Then w must have two neighbors u and u that are in T eo since v must be distance two from a least two vertices in T eo. However, this implies that N T (w) T eo 3, which is a contradiction. We now make the connection between every-other trees and 0-eigenvectors of A(T ). Theorem Given a tree T and an every-other tree T eo of T, a standard representation vector x of T eo is a 0-eigenvector of A(T ). Proof. Assume v is a vertex of T. If v is not adjacent to any vertex in T eo then (Ax)(v) = 0 by the definitions of A and x (note that this covers the case where v T eo ). If v is adjacent to u, w in T eo then (Ax)(v) = = 0. Therefore Ax = 0. We may also decompose arbitrary 0-eigenvectors of A(T ) using every-other trees. Observation Given a λ-eigenvector x of A(G), x(v) = λx(u). v u Proposition Given a tree T and 0-eigenvector x of A(T ), 1. if u and w are adjacent in T, then x(u) = 0 or x(w) = 0, and 2. T has a leaf v such that x(v) 0. Proof.

22 15 1. Assume not. Then by Observation 2.2.4, w has a neighbor v 1 u such that x(v 1 ) 0. Similarly, v 1 has a neighbor v 2 w such that x(v 2 ) 0. Proceeding in this fashion we derive an infinite sequence {v 1, v 2,... } such that x(v i ) 0. As T is a tree, each vertex in the sequence is unique and we reach a contradiction. 2. Assume not. Then there exists a vertex v 1 such that v 1 is not a leaf and x(v 1 ) 0. Now none of the neighbors of v 1 are leaves else we have a contradiction to Observation Let w 1 be a neighbor of v 1. Then, by Observation 2.2.4, w 1 has a neighbor v 2 other than v 1 such that x(v 2 ) 0. Similarly, choose w 2 a neighbor of v 2 other than w 1 (possible as deg v 2 2) and w 2 must have a nonzero neighbor v 3 v 2. Proceeding in this fashion we derive an infinite sequence {v 1, v 2,... } such that x(v i ) 0 because no leaf l has x(l) 0. However, as T is a tree, each vertex in the sequence is unique and we reach a contradiction. The preceding proposition is also a result of [6]. Note that the preceding implies that every 0-eigenvector of A(T ), T a tree, has at least two nonzero leaves. This proposition gives rise to Algorithm 1. The fact that the algorithm produces an every-other tree from an arbitrary 0-eigenvector of A(T ) is straightforward. Theorem For T a tree, if y is a 0-eigenvector of A(T ), then y = α 1 y α k y k where y i is a standard representation vector for an every-other tree T (i) eo of T. Proof. Apply Algorithm 1 to y to find an every-other tree T (1) eo in the support of y. Let α 1 = min (1) v T {y(v)} and v 1 some vertex in T eo (1) such that y(v 1 ) = α 1. Let y 1 be a standard eo representation vector of T eo (1) such that y 1 (v 1 ) = 1. Then y α 1 y 1 is a 0-eigenvector for A(T ) and support of y α 1 y 1 is smaller than the support of y. Apply Algorithm 1 to y α 1 y 1 to obtain T (2) eo, α 2, y 2 and v 2. Repeat the process until the support of y has been exhausted. Then y = α 1 y α k y k. Corollary Given a tree T, if k is the dimension of the null space of A(T ) then there exists a set of k every-other trees {T (1) eo,..., T (k) eo } such that the set of standard representation vectors for T (i) eo is linearly independent.

23 16 Algorithm 1 Algorithm for finding every-other trees in 0-eigenvectors of A(T ) Input: y, a 0-eigenvector for A(T ) and l, a leaf of T in the support of y Output: S, an every-other tree of T contained in the support of y and containing l S = {l} NextVertices = {l} stop = False RequiredLeaves = 2 while not stop do Vertices = NextVertices NextVertices = {} for v Vertices do RequiredLeaves = RequiredLeaves + max{0, N T (v) 2} for w in N T (v) \ S do Proposition guarantees that y(w) = 0 There is a u w such that sign(y(u)) = sign(y(v)) Observation NextVertices = NextVertices {u} end for end for S = S NextVertices if number of leaves in S = RequiredLeaves then stop = True end if end while

24 17 Proof. Assume y 1,..., y k is a basis for the null space of A(T ) and y i is a linear combination of y i1,..., y iki with y ij a standard representation vector of an every-other tree. Then {y ij } spans the null space of A(T ). Thus there exists a basis for A(T ) that is a subset of {y ij }. We now make the connection to the normalized Laplacian by use of harmonic eigenvectors [4]. If y is an eigenvector of L for λ, then z = D 1/2 y is a harmonic λ-eigenvector of L. Observation A vector z is a harmonic λ-eigenvector of L if and only if z is a λ- eigenvector of D 1 L. Further, given a harmonic λ-eigenvector z of L, z(v) = (1 λ)z(u)d(u). v u Thus, a vector is a 0-eigenvector of A if and only if it is a harmonic 1-eigenvector of L. Proposition Given a tree T, T has a set of k every-other trees whose standard representation vectors form a linearly independent set if and only if the multiplicity of 1 as an eigenvalue of L(T ) is at least k. Proof. Assume T has a set of k every-other trees whose standard representation vectors form a linearly independent set. This set induces k linearly independent 0-eigenvectors for A(T ) by Theorem This set is also a set of k linearly independent harmonic 1-eigenvectors for L(T ) which induces a set of k linearly independent 1-eigenvectors for L(T ). Now assume the multiplicity of 1 as an eigenvalue of L(T ) is at least k. Thus, the multiplicity of 0 as an eigenvalue of A(T ) is at least k. Thus by Corollary 2.2.7, T has a set of at least k every-other trees whose standard representation vectors form a linearly independent set. 2.3 An example of an infinite family of cospectral pairs of trees Now that we have established properties for lower bounding the multiplicity of 1 in the the spectrum of L, we can use this to establish a pair of trees as cospectral with respect to the normalized Laplacian. We begin with some useful propositions. Proposition If a P 7 (as enumerated in Figure 2.2) is an induced subgraph of a given graph G so that vertex 7 is the only vertex in P 7 with any neighbors in G P 7 (i.e. N G (i) P 7

25 18 for i = 1,..., 6), then p(t) = t 2 2t + 1/4 is a factor of p L (t), the characteristic polynomial of L(G). Further, L(G) has eigenvectors for the roots of p(t) with support contained in P Figure 2.2 P 7 Proof. Let G be a graph with P 7 an induced subgraph as described in Proposition Enumerate the vertices of G such that the P 7 is labeled as in Figure 2.2 with the remaining vertices labeled arbitrarily. Let λ > µ be the roots of p(t). Consider the vectors x and y with entries indexed by the vertices of G. Let 3 i = 1, 3 i = 1, x i = 1 i = 2, 3, 3 i = 4, y i = 1 i = 2, 3, 3 i = 4, 1 i = 5, 6, 0 else, 1 i = 5, 6, 0 else, 1 if i = j, and L D = L(G)D(G) 1. Thus, (L D ) ij = 1/ deg j if i j, and 0 else. Then a routine computation gives L D x = λx and L D y = µy. We conclude the proof by noting that L L D as D 1/2 LD 1/2 = L D. Proposition If the graph W (see Figure 2.3) is an induced subgraph of a given graph G so that vertex 9 is the only vertex in W with any neighbors in G W (i.e. N G (i) W for i = 1,..., 8), then p(t) = t 2 2t + 1/4 is a factor of the characteristic polynomial of L(G), p L(G) (t). Proof. Let G be a graph with W an induced subgraph as described in Proposition Enumerate the vertices of G such that the W is labeled as in Figure 2.3 with the remaining

26 Figure 2.3 W vertices labeled arbitrarily. Let λ > µ be the roots of p(t). Consider the vectors x and y with entries indexed by the vertices of G. Let 2 3 i = 1, 2 3 i = 1, x i = 1 i = 2, 3, 4, 2 3 i = 5, y i = 1 i = 2, 3, 4, 2 3 i = 5, 1 i = 6, 7, 8, 1 i = 6, 7, 8, 0 else, 0 else, and L D = L(G)D(G) 1. Then a routine computation gives L D x = λx and L D y = µy. The result holds as L(G) L D. Definition Define the sequence of graphs G k as follows: begin with the graph G 1 (Figure 2.4) on vertices. Figure 2.4 G 1 To construct G k, a graph on 3 + 4k vertices, attach k 1 additional P 3 s and leaves to the vertex with degree > 2 (call this vertex v hd ). Note that deg v hd = 2(k + 1). Enumerate the vertices in the following fashion: let 1, 2,..., k + 2 be the leaves that are adjacent to v hd and k + 3j for j = 1, 2,..., k be the non-leaf neighbors of v hd. Then let k + 3j + 1 be the degree 2 neighbor of k + 3j and k + 3j + 2 the degree 1 neighbor of k + 3j + 1. Finally, the vertex

27 k is v hd. See Figures 2.5 and 2.6 for G 2 (which is shown with enumerated vertices) and G 3 respectively Figure 2.5 G 2 Figure 2.6 G 3 Proposition p k (t) = t 2 2t + 2k+1 4k+4 is a factor of p L(G k )(t). Proof. The following is a construction of the eigenvectors of L D := L(G k )D(G k ) 1 for λ > µ, the roots of p k (t). Consider the vectors x and y with entries indexed by the vertices of G k. Let k i = 1, 2,..., k + 2, k i = 1, 2,..., k + 2, x i = 1 i = k + 3j, a k i = k + 3j + 1, y i = 1 i = k + 3j, a k i = k + 3j + 1, k + 1 i = k + 3j + 2, k + 1 i = k + 3j + 2, ka k i = 3 + 4k, ka k i = 3 + 4k, for j = 1, 2,..., k and a k = (2k + 3)(k + 1). Clearly the roots of p k (t) are For i = 1, 2,..., k + 2, 2k ± 4(k + 1) = 1 ± a k deg v hd (L D x) i = k ka k deg v hd = kλ = (λx) i as i is only adjacent to v hd. For i = k + 3j for j = 1, 2,..., k, (L D x) i = a k ka k deg v hd = 1 + (λ 1)(k + 1) k(λ 1) = λ = (λx) i

28 21 as i is adjacent to i + 1 (the middle of a P 3 ) and v hd. For i = k + 3j + 1 for j = 1, 2,..., k, (L D x) i = 1 2 a k (k + 1) = a k λ = (λx) i since i is adjacent to i 1, a non-leaf neighbor of v hd and i + 1, a leaf which is not adjacent to v hd. For i = k + 3j + 2 with j = 1, 2,..., k, (L D x) i = a k 2 + (k + 1) = deg v hd(λ 1) + deg v hd 2 = λ deg v hd 2 = (k + 1)λ = (λx) i as i is only adjacent to i 1, the middle of a P 3. Lastly, for i = 3 + 4k, (L D x) i = k(k + 2) k ( ) ka k = k 2 + a k + (k + 1) = ka k λ = (λx) i since i is adjacent to k + 2 leaves, which have value k in x, and k degree 2 vertices, which have value 1 in x. Thus, we have L D x = λx. Analogous computations show that L D y = µy. Definition Define the sequence of graphs H k as follows: begin with the graph H 1 G 1 on vertices with the vertices labeled as shown in Figure v hd 1 4 Figure 2.7 H 1 To construct H k, a graph on 3+4k vertices, add k 1 additional copies of H 1 [{1, 2, 3, 4}] and add an edge between v hd and vertex 1 in each copy of H 1 [{1, 2, 3, 4}]. Note that deg v hd = k +1. Enumerate the vertices in the following fashion: let 1 + 4j for j = 0, 1,..., k 1 be the degree 4 neighbors of v hd and let 2 + 4j, 3 + 4j, and 4 + 4j be the neighbors of 1 + 4j. Finally, let 4k + 1 be the remaining leaf in G k, 4k + 2 the degree 2 vertex and 4k + 3 be v hd. See Figures 2.8 and 2.9 for H 2 (which is shown with enumerated vertices) and H 3 respectively.

29 Figure 2.8 H 2 Figure 2.9 H 3 Proposition p k (t) = t 2 2t + 2k+1 4k+4 is a factor of p L(H k )(t). Proof. The following is a construction of the eigenvectors of L D := L(H k )D(H k ) 1 for λ > µ, the roots of p k (t). Consider the vectors x and y with entries indexed by the vertices of H k. Let b k i = 1 + 4j, b k i = 1 + 4j, x i = 1 i = 2 + 4j, 3 + 4j, 4 + 4j, 2k i = 4k + 1, y i = 1 i = 2 + 4j, 3 + 4j, 4 + 4j, 2k i = 4k + 1, kb k i = 4k + 2, kb k i = 4k + 2, k i = 4k + 3, k i = 4k + 3, for j = 0, 1,..., k 1 and b k = j = 0, 1,..., k 1, 4(2k+3) k+1. Thus, λ = b k, µ = b k. For i = 1 + 4j with (L D x) i = b k + 3 k k + 1 = b k + 2k + 3 k + 1 = b k b k 2 = (λx) i as i is adjacent to i + 1, i + 2, i + 3, and 4k + 3. For i = 2 + 4j, 3 + 4j, or 4 + 4j with j = 0, 1,..., k 1, as i is only adjacent to 1 + 4j. For i = 4k + 1, (L D x) i = 1 4 b k 1 = λ = (λx) i (L D x) i = 2k + k ( 2 b k = 2k ) 4 b k = 2kλ = (λx) i

30 23 as i is only adjacent to i + 1. For i = 4k + 2, (L D x) i = 2k kb k k ( k + 1 = k b k + 2k + 3 ) = k (b k + 14 ) k + 1 b k 2 = kb k λ = (λx) i as i is adjacent to i 1 and i + 1. For i = 4k + 3, (L D x) i = k 4 b k + k ( 2 b k + k = k ) 4 b k = kλ = (λx) i as i is adjacent to 1+4j with j = 0, 1,..., k 1 and to i 1. Thus, we have L D x = λx. Similar computations show that L D y = µy. We conclude by noting that L(H k ) L D. Theorem For all positive integers k, G k and H k have the same spectrum (including multiplicity) with respect to the normalized Laplacian. Proof. Consider the graphs G k and H k for a given k. Both G k and H k are bipartite graphs, so t and t 2 are factors of both p L(Gk )(t) and p L(Hk )(t). The graph G k has a set of 2k + 1 every-other trees whose standard representation vectors form a linearly independent set: {1, 2} and {1, j + 2}, {1, k + 3j, k + 3j + 2} for j = 1, 2,..., k. Thus (t 1) 2k+1 is a factor of p L(Gk )(t) by Proposition Similarly, H k also has 2k + 1 every-other trees whose standard representation vectors form a linearly independent set: {2, 6,..., 2 + 4(k 1)} {4k + 1, 4k + 3} and {2 + 4j, 3 + 4j}, {2 + 4j, 4 + 4j} for j = 0, 1,..., k 1. Hence, (t 1) 2k+1 is a factor of p L(Hk )(t) as well. The graph G k has k 1 copies of the P 7 described in Proposition beginning at vertex k +5 and ending at vertex 5+k +3j for j = 1,..., k 1. These induce k 1 linearly independent eigenvectors of the eigenvalues described in Proposition 2.3.1, so (t 2 2t + 1/4) k 1 is a factor of p L(Gk )(t). Similarly, H k has k 1 copies of the graph W as described in Proposition containing the vertices 1, 2, 3, 4, 3 + 4k and 4j + 1, 4j + 2, 4j + 3, 4j + 4 for j = 1,..., k 1, so (t 2 2t+1/4) k 1 is a factor of p L(Hk )(t) as well. Finally, Propositions and guarantee that p k (t) is a factor of both p L(Gk )(t) and p L(Hk )(t). This is the complete factorization of p L(Gk )(t) and p L(Hk )(t) as deg(p L(Gk )(t)) = deg(p L(Hk )(t)) = 4k + 3. Hence ( p L(Gk )(t) = p L(Hk )(t) = t(t 2)(t 1) 2k+1 t 2 2t + 1 ) k 1 ( t 2 2t + 2k + 1 ). 4 4k + 4

31 Using the weighted normalized Laplacian A weighted graph is a graph in which each edge is assigned a positive value and each nonedge is assigned 0. The degree of a vertex is the sum of its incident edge weights. We define the adjacency matrix A(G) = (a ij ) for a weighted graph G with vertices {v 1, v 2,..., v n }, edges e ij = {v i, v j }, and edge weights w ij by a ij = w ij. The normalized Laplacian matrix associated with a weighted graph G is L(G) := I D 1/2 A(G)D 1/2 where D = diag(deg v 1, deg v 2,..., deg v n ). The definition of harmonic eigenvectors of L(G) is analogous to the earlier definition with nonweighted graphs, i.e. if y is an eigenvector of L(G) for λ, then z = D 1/2 (G)y is a harmonic λ-eigenvector of L(G). Note that if G is an unweighted graph and G is a weighted graph with the same vertices and edges as G in which each edge is given weight 1, then A(G) = A(G) and L(G) = L(G). Proposition Assume G is a graph, v 1,..., v k are pairwise twins for some k 1 and G is G where each edge is given weight 1. Then p L(G) (t) = (t 1) k 1 p L(G )(t) where G is G with v 1,..., v k 1 removed and the weights of the edges incident with v k multiplied by k. Proof. It is a straight forward verification to show that the vector x (i) with 1 if j = k x (i) (v j ) = 1 if j = i 0 else is a 1-eigenvector of L(G) for i = 1,..., k 1. We may form the remaining eigenvectors of L(G), y (l) for eigenvalue µ l, so that y (l) x (j) for all l and j. Hence, y (l) (v j ) = c l for j = 1,..., k for some constant c l. Consider the harmonic eigenvectors which correspond to the vectors w (l) R n k+1 where w (l) (v j ) = y (l) (v j ) for j = k,..., n, i.e., D 1/2 w (l). Since deg G v = deg G v, Observation gives L(G )w (l) = µ l w (l). The result follows.

32 Characterization of diameter 3 trees Using this knowledge of the weighted normalized Laplacian we may classify trees which are cospectral with diameter three trees with respect to the normalized Laplacian. The weighted path (a 1, a 2,..., a n 1 ) is the weighted graph with vertices {v 1,..., v n }, edges {e 12, e 23,..., e n 1 n } and edges weights {w 12 = a 1,..., w n 1 n = a n 1 }. Proposition The characteristic polynomial of the weighted path (m, 1, n m 2) is ( ) p 1 (t) = t(t 2) t 2 n 1 2t + (n m 1)(m + 1) and the characteristic polynomial of the weighted path (m, 1, 1, n m 3) is ( ) p 2 (t) = t(t 2)(t 1) t 2 n 1 2t +. 2(n m 2)(m + 1) Proof. Via the similarity L(G) L(G)D 1 (G), we have L((m, 1, n m 2)) which has the characteristic polynomial p 1 (t) and 1 m m m n m m n+2 m n+1 1 L((m, 1, 1, n m 3)) 1 m m m m n m n+3 m n+2 1 which has the characteristic polynomial p 2 (t).

33 26 Proposition If T 1 is a tree with diameter 3 on n vertices (i.e. of the form in Figure 2.10), then p L(T1 )(t) = t(t 2)(t 1) n 4 (t 2 2t + for some 1 m n/2 1. ) n 1 (n m 1)(m + 1) 2. If T 2 is a tree with diameter 4 on n vertices and of the form in Figure 2.11, then ) p L(T2 )(t) = t(t 2)(t 1) (t n 4 2 n 1 2t + 2(n m 2)(m + 1) for some 1 m n/2 1. Proof 1. Since T 1 is a diameter 3 tree, then it has the structure shown in Figure 2.10 for some 1 m 1 n/2 1. Thus by Proposition 2.4.1, T 1 has the characteristic polynomial (t 1) n 4 p L(T 1 )(t) where T 1 from Proposition is the weighted path (m, 1, n m 2). The result follows 2. T 2 has the structure shown in Figure 2.11 for some 1 m n/2 1. Thus by Proposition 2.4.1, T 2 has the characteristic polynomial (t 1) n 5 p L(T 2 )(t) where T 2 is the weighted path (m, 1, 1, n m 3). The result follows from Proposition m+1 1 m+1 2 m+2 2 m+2 n-1 n n-2 n-1 n m n-2 m n-3 Figure 2.10 T 1 Figure 2.11 T 2 Corollary The following are consequences of Proposition The diameter 3 tree on 7k + 5 vertices with degree sequence (2 + 4k, 3 + 3k, 1,..., 1) is cospectral with the diameter 4 tree of the form T 2 on 7k + 5 vertices with degree sequence (3 + 6k, k + 1, 2, 1,..., 1).

34 27 2. The diameter 3 tree on 7k + 6 vertices with degree sequence (4 + 4k, 2 + 3k, 1,..., 1) is cospectral with the diameter 4 tree of the form T 2 on 7k + 6 vertices with degree sequence (4 + 6k, k + 1, 2, 1,..., 1). Figures 2.12 and 2.13 are cospectral graphs by Corollary case 1, and Figures 2.14 and 2.15 are cospectral graphs by Corollary case 2. Figure 2.12 T 1 Figure 2.13 T2 Figure 2.14 T 1 Figure 2.15 T 2 Theorem No two distinct diameter 3 trees are cospectral. Proof. Assume S 1 and S 2 are cospectral diameter 3 trees. Without loss of generality, by Proposition 2.4.3, ) p L(S1 )(t) = t(t 2)(t 1) (t n 4 2 n 1 2t (n m 1 1)(m 1 + 1) ) p L(S2 )(t) = t(t 2)(t 1) (t n 4 2 n 1 2t (n m 2 1)(m 2 + 1) for 1 m 1 m 2 n/2 1. As S 1 and S 2 are cospectral, (n m 1 1)(m 1 + 1) = (n m 2 1)(m 2 + 1). For purposes of contradiction, assume m 1 + k = m 2 for some k 1. Then (n m 2 + k 1)(m 2 k + 1) = (n m 2 1)(m 2 + 1) (n m 2 1)(m 2 k + 1) + k(m 2 k + 1) = (n m 2 1)(m 2 + 1) (n m 2 1)(m 2 + 1) + k(2m n k) = (n m 2 1)(m 2 + 1) k(2m n k) = 0 2(m 2 + 1) n k = 0 However, m n/2, thus 2(m 2 + 1) n 0. But k is strictly negative and we arrive at a contradiction. Therefore, m 1 = m 2 and S 1 and S 2 are isomorphic.

35 28 Theorem The only trees which are cospectral with a diameter 3 tree are of the form T 2 in Figure Proof. Assume that T 1 is a diameter 3 tree on n vertices and T is a tree cospectral with T 1. The multiplicity of 1 as an eigenvalue of L(T 1 ) is n 4 by Proposition Thus T must have diameter greater than 3 by Theorem Trees with diameter greater than 4 have at least 6 distinct eigenvalues, thus the multiplicity of 1 is at most n 5. Therefore T has diameter 4. Let v be the vertex in T such that d T (v, u) 2 for all u T. Case 1: v has a leaf as a neighbor. No every-other tree in T contains v. Thus if T eo is an every-other tree in T, T eo = {v 1, v 2 } for v 1, v 2 leaves of the same vertex. There are are most n 6 every-other trees of this form whose standard representations form a linearly independent set of vectors. Thus, the multiplicity of 1 is at most n 6. Case 2: v does not have a leaf as a neighbor. Assume that v 1, v 2,..., v l are the neighbors of v and v i1, v i2,..., v iki are the neighbors of v i. Let T (0) eo = {v, v 11, v 21,..., v l1 } and T (ij) eo = {v i1, v ij } for i = 1,..., l and j = 2,..., k i. Let y 0 be the standard representation vector of T (0) eo y 0 (v) = 1 and y ij be the standard representation vector of T (ij) eo with y ij (v i1 ) = 1. The set I = {y 0 } {y ij : i = 1,..., l, j = 2,..., k l } is linearly independent. Assume T eo is an every-other tree of T. If T eo does not contain v, then T eo = {v ij, v ij } for some i, j, j. Thus, any standard representation vector of T eo is the linear combination of y ij and y ij. If T eo does contain v then T eo = {v, v 1j1, v 2j2,..., v ljl } for some j 1,..., j l. Thus, any standard representation vector of T eo is the linear combination of y 0, y 1j1,..., y ljl. Thus I is a basis for the space of standard representation vectors of all every-other trees of v. By construction, I = n 2l. Therefore, the multiplicity of 1 as an eigenvector of L(T ) is at most n 2l by Corollary so l = 2, i.e. T T 2. There are diameter three trees which are cospectral with non-trees. See Figure 2.16 for with the smallest example. The non-tree graphs in Figure 2.16 give rise to another interesting application of the weighted normalized Laplacian.

36 29 Figure 2.16 A cospectral family Balanced two-weighted paths It is well known that the eigenvalues of the adjacency matrix of a cycle on n vertices, C n, are λ k = ω k +ω (n 1)k for k = 0, 1,..., n 1 with ω = e 2πi/n. The eigenvector for λ k can be described as v (k) = (1, ω k, ω 2k,..., ω (n 1)k ) T. Now, as C n is 2-regular, L(C n )v (k) = ( λ k) v (k). Define µ k = λ k. Proposition Assume that n = 2l and ˆv (k) is a vector of length l + 1 with 2 i = 1 ˆv (k) i = ω (i 1)k + ω (n i+1)k i = 2,..., l ( 1) k 2 i = l + 1 Then L(P l+1 )ˆv (k) = µ k ˆv (k) where P l+1 is the path on l + 1 vertices.

37 30 Proof. For i = 1 : ( L(P l+1 )ˆv (k)) 1 = ( ω k + ω (n 1)k) = 2µ k = µ kˆv (k) 1, 2 : ( L(P l+1 )ˆv (k)) 2 = 1 + ωk + ω (n 1)k 1 2 ω2k 1 2 ω(n 2)k = µ kˆv (k) 2, 3,..., l 1 : ( L(P l+1 )ˆv (k)) = 1 i 2 ˆv(k) i 1 + ˆv(k) i 1 2 ˆv(k) i+1 = 1 2 ( ω (i 2)k + ω (n i+2)k) + ω (i 1)k + ω (n i+1)k 1 ( ω ik + ω (n i)k) = µ kˆv (k) i, 2 l : l + 1 : ( L(P l+1 )ˆv (k)) ( = 1 ω (l 2)k + ω (l+2)k) + ω (l 1)k + ω (l+1)k ( 1) k = µ kˆv (k) l 2 l, ( L(P l+1 )ˆv (k)) l+1 = 1 2 (ω (l 1)k + ω (l+1)k) + 2( 1) k = 2 ( 1 2 ( 1)k ω (n 1)k 1 2 ( 1)k ω k + ( 1) k ) = ( 1) k 2µ k = µ kˆv (k) l+1. The result follows. Proposition If ˆv (k) 1, 2,..., l/2. is defined as in Proposition 2.4.7, ˆv (k) i (k) = ( 1) kˆv l i+2 for i = Proof. This is clear for i = 1. Thus if i > 1, ˆv (k) l i+2 = ω(l i+1)k + ω (l+i 1)k = ω (n i+1)k+lk + ω (i 1)k+lk (k) = ( 1) kˆv i. Definition For a, b > 0, the balanced two-weighted path P s,a,b is the weighted path on 2s + 1 vertices, (a, a,..., a, b, b,..., b), such that a and b both appear s times. We will now show that weighting the path in this manner does not affect the spectrum.

38 31 Proposition Assume l = 2s and v (k) is a vector of length l + 1 with v (k) i = ˆv (k) i i = 1,..., s a+1 2a ˆv (k) i i = s + 1 for k even and v (k) i = 1 a ˆv (k) i i = s + 2,..., 2s + 1 ˆv (k) i i = 1,..., s a+1 2a ˆv (k) i i = s + 1 for k odd where ˆv (k) i aˆv (k) i i = s + 2,..., 2s + 1 is defined as in Proposition Then L(P s,a,1 ) v (k) = µ k v (k) for all k = 0, 1,..., l. Hence, P s,a,1 and P 2s+1 are cospectral. Proof. By the normalization, (L(P s,a,1 )) ij = (L(P 2s+1 )) ij for all pairs (i, j) / {(s, s + 1), (s + 1, s), (s + 1, s + 2), (s + 2, s + 1)}. Assume k is even. Then ( L(P s,a,1 ) v (k)) i = ( L(P s,a,1 )ˆv (k)) i = µ kˆv (k) i 1. And ( L(P s,a,1 ) v (k)) i = 1 a ( L(Ps,a,1 )ˆv (k)) i = 1 a µ kˆv (k) i Therefore, we only need to consider i = s, s + 1, s + 2. For i = s, ( L(P s,a,1 ) v (k)) a = 1 s 2 v(k) s 1 + v(k) s v (k) s+1 = 1 2(a + 1) 2 ˆv(k) For i = s + 1, (Note: below we use the fact that ˆv (k) s = µ k v (k) i for i = 1,..., s = µ k v (k) i for i = s + 3,..., 2s + 1. s 1 + ˆv(k) s 1 2 ˆv(k) s+1 = µ kˆv s (k) = µ k v s (k). = ˆv (k) s+2 from Proposition 2.4.8) ( L(Ps,a,1 ) v (k)) a = v s+1 s (k) 2(a + 1) + v (k) s+1 1 2(a + 1) v (k) s+2 a a + 1 = ˆv s (k) + ˆv (k) s+1 1 ˆv (k) s+2 2(a + 1) 2a 2a(a + 1) = a + 1 ( ˆv s (k) + ˆv (k) s+1 ) 2a = a + 1 2a µ kˆv (k) s+1 = µ k v (k) s+1.

39 32 For i = s + 2, ( L(P s,a,1 ) v (k)) = 1 v (k) s+1 + s+2 v(k) s+2 1 2(a + 1) 2 v(k) s+3 = 1 µ kˆv (k) a s+2 = µ k v (k) s+2. Now assume k is odd. As before, we only need to consider the cases where i = s, s + 1, s + 2. For i = s, the calculations are the same as when k was even. For i = s + 1, using the fact that ˆv (k) s = ˆv (k) s+2, ( L(P s,a,1 ) v (k)) s+1 = a ) ( ˆv s (k) + ˆv (k) s+1 ˆv(k) s+2 = µ k v (k) s+1. 2(a + 1) Finally, for i = s + 2, ( L(P s,a,1 ) v (k)) = 1 ( 1 s+2 a 2 ˆv(k) s+1 + ˆv(k) s+2 1 ) 2 ˆv(k) s+3 = 1 µ kˆv (k) a s+2 = µ k v (k) s+2. Therefore, L(P s,a,1 ) v (k) = µ k v (k). Given a weighted graph, scaling all the edge weights by a fixed amount does not change the spectrum of the normalized Laplacian. Thus we get the following corollary. Corollary P s,a,b and P 2s+1 are cospectral for the normalized Laplacian for any a, b N An example of cospectral bipartite graphs We will now examine an application of using two-weighted paths to determine the spectrum of a simple graph. Definition Consider the path P 2s+1 with vertices v 1,..., v 2s+1 and edges e 12,..., e 2s 2s+1. The graph D s,a,b on n := (a + b + 2)s + 1 vertices is the graph P 2s+1 in which a 1 twins of v s 2k for k = 0, 1,..., s/2 have been added and b 1 twins of v s+2k for k = 1, 2,..., s/2 have been added. Theorem D s,a,b has the characteristic polynomial p L(Ds,a,b )(t) = (t 1) n (2s+1) p L(P2s+1 )(t). Proof. Apply Proposition to D s,a,b iteratively until no twins remain. The resulting graph is P s,a,b. Corollary completes the proof.