Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph Leslie Hogben Department of Mathematics, Iowa State University, Ames, IA Directions in Combinatorial Matrix Theory Banff International Research Station May 8, 2004

Definitions and Notation A graph G = (V, E) means a simple undirected graph (no loops, no multi-edges), vertices V  Z +. The order is the number of vertices. The degree of vertex k, deg k, is the number of edges incident with k. A tree T is a connected graph with no cycles. P n is a path on n vertices C n is a cycle on n vertices K n is the complete graph on n vertices. K 1,n is a star on n+1 vertices W n is a wheel on n vertices G – v is the graph resulting from the deletion of vertex v and all its incident edges from G. The induced subgraph is the result of deleting all vertices not S from G. G is regular of degree r if every vertex has degree r. Unless otherwise noted, a graph is assumed connected, because each connected component can be analyzed separately.

Spectral Graph Theory Spectral graph theory uses the spectra of matrices associated with the graph, such as the adjacency matrix, the Laplacian matrix, or the normalized Laplacian, to provide information about the graph. One goal is to characterize a graph or obtain information about the graph from the spectra of these matrices. There also important applications to other fields such as chemistry. Hundreds of papers and several books such as Spectra of Graphs by Cvetkovic, Doob, and Sachs, Recent Results in the Theory of Graph Spectra by Cvetkovic, Doob, Gurman, Torgasev, and Spectral Graph Theory by Chung have been written on spectral graph theory.

Let G be a graph with vertices {1,…,n}. Matrices associated with G: Let A denote the adjacency matrix of G ( = 1 if {i,j} is an edge, else 0). Denote its ordered spectrum Let D denote the diagonal matrix whose ith entry is the degree of vertex i. The normalized adjacency matrix is with ordered spectrum Let A = D + A. Denote its ordered spectrum The normalized form of A is is an eigenvector for eigenvalue 1 of (and 2 of ). ( 1 denotes the vector all of whose entries are 1.)

Example

The following graph parameters are determined by the adjacency matrix A and its characteristic polynomial The first equality is obtained by viewing the coefficient of as the sum of the principal minors of order k, and the second is obtained by considering walks. Unfortunately these results do not extend cleanly to longer cycles, as can be seen by considering the 4-cycle. However, counting disjoint cycles can be used to evaluate the coefficients.

If two graphs have different spectra (equivalently, different characteristic polynomials) then they are not isomorphic. However, non-isomorphic graphs can be cospectral: Examples of spectrally determined graphs: Complete graphsEmpty graphs Graphs with one edgeGraphs missing only 1 edge Regular of degree 2 Regular of degree n-3 m K n K n,n,…,n However, “most” trees are not spectrally determined.

 (M) denotes the spectral radius of M. Perron-Frobenius Theorem Let M be an irreducible non-negative n  n matrix. Then a)  (M) > 0 b)  (M) is an eigenvalue of M c)  (M) is algebraically simple as an eigenvalue of M d)there is a positive vector x such that M x =  (M) x The matrices are all nonnegative (and irreducible if G is connected). The Perron root of M is the largest eigenvalue of M (for M = A it is called the index of G).

Additional matrices associated with G: L = D – A is the Laplacian of G The normalized Laplacian of G is For any graph If G is regular of degree r then

Example

The matrices are also connected via the incidence matrix. The (vertex-edge) incidence matrix N of graph G with n vertices and m edges is the n  m 0,1-matrix with rows indexed by the vertices of G and columns indexed by the edges of G such that the v,e entry of N is 1 (respectively, 0) if edge e is (respectively, is not) incident with vertex v. Then N N T = D + A = A. If G’ is any orientation of G and N’ is the oriented incidence matrix then N’ N’ T = D – A = L, and L = So are all positive semidefinite, and so have nonnegative eigenvalues.

The following facts are straightforward (if G is connected and not K n ). if and only if G is bipartite If G is not connected, the multiplicity of 0 as an eigenvalue of L is the number of connected components. For each of the matrices, the spectrum of is the union of the spectra of the components.

The second smallest eigenvalue of L,, is called the algebraic connectivity. The vertex connectivity,, is the minimum number of vertices in a cutset (for a graph that is not the complete graph). Theorem If G is not K n, the vertex connectivity is greater than or equal to the algebraic connectivity, i.e., Example

The distance between two vertices in a graph is the length of the shortest path between them. The diameter of a graph G, diam(G), is maximum distance between any two vertices of G. Theorem The diameter of G is less than the number of distinct eigenvalues of the adjacency matrix of G. There are also several other diameter results involving the Laplacian and normalized Laplacian.

Inverse Eigenvalue Problem of a Graph (IEPG) S n = the set of symmetric real n  n matrices For B  S n, the graph of B, G (B), is the graph with vertices {1,…,n} and edges E = { {i,j} | a ij  0}. 1 G (B) = For G a graph with vertices {1,…,n}, the set of symmetric matrices of the graph is S (G) = { B  S n | G (B) = G}. The Inverse Eigenvalue Problem of a Graph is to characterize the possible spectra of matrices in S (G).

Note that are all in S (G). L is a generalized Laplacian matrix of G if L has nonpositive off- diagonal elements and L  S (G). Note that in this case, -L has non-negative off-diagonal elements and there is a real number c such that cI - L is non-negative, so if G is connected, the least eigenvalue of L is simple. Theorem Let L be a generalized Laplacian matrix of the graph G. If G is 3-connected and planar then has multiplicity less than 4. Much recent work with generalized Laplacians is based on Colin de Verdière matrices. There interesting connections between the work on generalized Laplacians and the Inverse Eigenvalue Problem of a Graph.

Definitions and Notation Let B  S n.  (B) = {  1, …,  n } is the ordered spectrum of B (  k <  k+1 ). m B (  ) = the multiplicity of  as an eigenvalue of B The eigenvalue  is simple if m B (  ) = 1. M(G) = max{m B (  ) | B  S (G) } maximum multiplicity of G mr(G) = min{ rank B | B  S (G) } minimum rank of G M(G) + mr(G) = n. If H is an induced subgraph of G then mr(H) < mr(G). mr(K n ) = 1 and mr(P n ) = n - 1. Theorem [Fiedler] If m B (  ) = 1 for all    (B) for all B  S (G), then G = P n. Equivalently, mr(G) = n - 1 implies G = P n.

Theorem [Barrett and Loewy] mr(G) = 2 if and only if G is not K n and does not contain as an induced subgraph any of: P 4, K 3,3,3 = complete tripartite graph, or

Most of the progress on the Inverse Eigenvalue Problem of a Graph is for trees. Recall that for any graph, the diameter of G is less than the number of distinct eigenvalues of A, and the proof extends to show diam(G) < the number of distinct eigenvalues of any non-negative matrix B  S (G). If T is a tree and B  S (T), it is possible to find a 1,-1 diagonal matrix S with non-negative. Theorem [Johnson and Leal Duarte] If T is a tree, for any B  S (T), the diameter of T is less than the number of distinct eigenvalues of B. Thus, diam(T) < mr(T)

If B is an n  n matrix, B(k) is the (n -1)  (n -1) matrix obtained from B by deleting row and column k. If B  S (G), then B(k)  S (G-k). Interlacing Theorem Let B  S n, k  {1,…,n}. If the eigenvalues of B are  1 <  2 < … <  n and the eigenvalues of B(k) are  1 <  2 < … <  n-1, then  1 <  1 <  2 <  2 <  3 < … <  n-1 <  n. Corollary m B(k) (  )  {m B (  ) -1, m B (  ), m B (  ) +1}.

k is a Parter-Wiener (PW) vertex of B for eigenvalue  if m B(k) (  ) = m B (  ) + 1. k is a strong PW vertex of B for  if k is a PW vertex of B for  and  is an eigenvalue of at least three components of B(k). Parter-Wiener Theorem If T is a tree, B  S (T) and m B (  ) > 2, then there is a strong PW vertex of B for . Corollary If T is a tree, B  S (T), and  (B) = (  1, …,  n ) with  k <  k+1 then  1 and  n are simple eigenvalues H(G) = {k  V(G) | deg k > 3} is called the set of high degree vertices of G. Only high degree vertices can be strong PW vertices.

Example The star on n+1 vertices K 1,n has only one high degree vertex, say 1, so this vertex must be PW for any multiple eigenvalue of B with G(B) = K 1,n. By choosing the diagonal elements of B for 2,…,n to be 0, we obtain m B(k) (0) = n and so m B (0) = n -1 and mr(K 1,n ) = 2. (In this case the other two eigenvalues are necessarily simple). The Parter-Wiener Theorem need not be true for graphs that are not trees. Example For A the adjacency matrix of C 4, m A (0) = 2 but there is no PW vertex since C 4 - k is P 3 for any vertex k.

P(G), the path cover number of G, is the minimum number of vertex disjoint paths occurring as induced subgraphs of G that cover all the vertices of G.  (G) = max{p-q | there is a set of q vertices whose deletion leaves p paths} Theorem [Johnson and Leal Duarte] Let T be a tree. Then M(T) = P(T) =  (T). For any graph G,  (G) < M  G). [Barioli, Fallat, Hogben] For any graph G,  (G) < P  G).

Theorem [Johnson, Leal Duarte, Saiago] The possible ordered multiplicity lists of the following families of trees have been determined. Each possible list can be realized for any list of real numbers in order. Paths Stars Double Paths Generalized Stars Double Generalized Stars

However, there exist trees for which an ordered multiplicity list is possible but not attainable for all such ordered real number lists. Example [Barioli and Fallat]

A  S (BF). so A has ordered multiplicity list 1, 2, 4, 2, 1 but if B  S (BF) has the five distinct eigenvalues

We now examine matrices realizing minimum rank and having some special form such as A, a 0,1 matrix in S (G), or a generalized Laplacian of G. Note that finding a matrix in S (G) with non-negative off-diagonal elements and minimum rank is equivalent to finding a generalized Laplacian of minimum rank. Theorem [Hogben] If T is a tree and A is its adjacency matrix then there exists a 0,1 diagonal matrix D such that m 0 (A+ D) = M(T), and thus rank (A+D) = mr(T).

Theorem If T is a tree and A is its adjacency matrix then there exists a 0,1 diagonal matrix D such that m 0 (A+ D) = M(T), and thus rank (A+D) = mr(T). Proof: There exists a set Q of q vertices such that T – Q consists of p disjoint paths and p – q = M(T). For each path, remove alternate interior vertices so that the result is isolated vertices (and one path of 2 vertices if the path had an even number of vertices originally). Let Q’ be the set of q’ vertices consisting of the original q vertices and the additional alternate interior vertices deleted. Then Q’ has the property that T – Q consists of p’ disjoint paths of 1 or 2 vertices and p’ – q’ = M(T). Chose the diagonal of D to be 0 for isolated vertices and 1 for vertices in a path of 2 vertices in T – Q’. Then 0 is an eigenvalue of each of the p’ paths and, by interlacing, m 0 (A+D) = M(T). rank (A+D) = n – m 0 (A+D) = n – M(T) = mr(T).

By using the algorithm of Johnson and Saiago for producing the set Q of vertices to delete to obtain , we obtain the following algorithm for producing a 0,1 matrix M in S (T). Let  T (v) = deg T (v) - deg H(T) (v). To start, set T’ = T and Q = . Repeat : 1) Q’ = {v |  T’ (v) > 2} 2) Set Q = Q  Q’ 3) Set T’ = T’ - Q’ Until Q’ = . In each path, remove alternate interior vertices and add these to Q. D = diagonal(d 1,…,d n ) where

Example The shaded vertices are in Q and the red vertices have diagonal entry assigned 1 (all other diagonal entries are 0). Removed 1st iteration Removed 2nd iteration Removed alternate interior Diagonal entry assigned 1

In fact, it is not true for all graphs G that there is a diagonal matrix D with with rank(A+D) = mr(G). Example H=

However, there is a matrix M  S (H) with rank(M) = 2 = mr(H) and all off-diagonal entries non- negative (so a scalar translation of M is non-negative). And L = -M is a generalized Laplacian of minimum rank. H =

Question: For every graph G: Is there a matrix M  S (G) with rank(M) = mr(G) and all off-diagonal entries non-negative? Equivalently, is there a generalized Laplacian matrix L of G with rank(L) = mr(G) ?