Matrix and Matrix and Iowa State University and American Institute of Mathematics ICART 2008 May 28, 2008
Inverse Eigenvalue Problem for a Graph () Problem for a Graph of minimum rank Specific matrices A, L, L, Â, L, L among A, L, L, Â, L, L Colin de Verdière Type Parameters Matrix and Problem: Recent Results Minimum rank graph catalogs
Matrix and Matrix Studies patterns of entries in a matrix rather than values In some applications, only the sign of the entry (or whether it is nonzero) is known, not the numerical value Uses graphs or digraphs to describe patterns Uses graph theory and combinatorics to obtain results about matrices Inverse Eigenvalue Problem of a Graph () associates a family of matrices to a graph and studies spectra
Algebraic Graph Uses algebra and linear algebra to obtain results about graphs or digraphs Groups used extensively in the study of graphs Spectral graph theory uses matrices and their eigenvalues are used to obtain information about graphs. Specific matrices: adjacency, Laplacian, signless Laplacian matrices Normalized versions of these matrices Colin de Verdière type parameters associate families of matrices to a graph but still use the matrices to obtain information about the graph Matrix and
Matrix and Connections between these two approaches have yielded results in both directions. Terminology and notation All matrices are real and symmetric Matrix B = [b ij ] σ(b) is the ordered spectrum (eigenvalues) of B, repeated according to multiplicity, in nondecreasing order All graphs are simple Graph G = (V, E)
Matrix and Inverse Eigenvalue Problem: What sets of real numbers β,..., β n are possible as the eigenvalues of a matrix satisfying given properties of a matrix? Inverse Eigenvalue Problem of a Graph (): For a given graph G, what eigenvalues are possible for a matrix B having nonzero off-diagonal entries determined by G?
The graph G(B) = (V, E) of n n matrix B is V = {,..., n}, E = {ij : b ij 0 and i j}. Diagonal of B is ignored. Example: B = 2 5 0 0 0 0 0 5 0 0 0 G(B) 2 The family of matrices described by G is 4 Matrix and S(G) = {B : B T = B and G(B) = G}.
Matrix and Tools for B is irreducible if and only if G(B) is connected Any (symmetric) matrix is permutation similar to a block diagonal matrix and the spectrum of B is the union of the spectra of these blocks The diagonal blocks correspond to the connected components of G(B) It is customary to assume a graph is connected (at least until we cut it up)
Matrix and B(i) is the principal submatrix obtained from B by deleting the i th row and column. Eigenvalue interlacing: If β β n are the eigenvalues of B, and γ γ n are the eigenvalues of B(i), then β γ β 2 γ n β n [Parter 69], [Wiener 84] If G(B) is a tree and mult B (β) 2, then there is k such that mult B(k) (β) = mult B (β) +
Ordered multiplicity lists If the distinct eigenvalues of B are β < < β r with multiplicities m,..., m r, then (m,..., m r ) is called the ordered multiplicity list of B. Determining the possible ordered multiplicity lists of matrices in S(G) is the ordered multiplicity list problem for G The of G can be solved by solving the ordered multiplicity list problem for G proving that if ordered multiplicity list (m,..., m r ) is possible, then for any real numbers γ < < γ r, there is B S(G) having eigenvalues γ,..., γ r with multiplicities m,..., m r. If this case, for G is equivalent to the ordered multiplicity list problem for G Matrix and
[Fiedler 69], [Johnson, Leal Duarte, Saiago 0], [Barioli, Fallat 05] The possible ordered multiplicity lists of matrices in S(T ) have been determined for the following families of trees T : paths double paths stars generalized stars double generalized stars For T in any of these families, for G is equivalent to the ordered multiplicity list problem for G Matrix and It was widely believed that the ordered multiplicity list problem was always equivalent to for trees.
Example [Barioli, Fallat 0] For the tree T BF Matrix and the ordered eigenvalue list for the adjacency matrix A is ( 5, 2, 2, 0, 0, 0, 0, 2, 2, 5), so the ordered multiplicity list (, 2, 4, 2, ) is possible. But if B S(T BF ) has the five distinct eigenvalues β < β 2 < β < β 4 < β 5 with ordered multiplicity list (, 2, 4, 2, ), then β + β 5 = β 2 + β 4.
Matrix and First step in solving is determining maximum possibility multiplicity of an eigenvalue. maximum multiplicitiy M(G) = minimum rank mr(g) = max mult B(β) B S(G) min rank(b) B S(G) M(G) is the maximum nullity of a matrix in S(G) M(G) + mr(g) = G The Problem for a Graph is to determine mr(g) for any graph G.
Matrix and Examples: Path: mr(p n ) = n. Complete graph: mr(k n ) =? 0... 0 0?... 0 0 0?... 0 0........ 0 0 0...? 0 0 0...? is nonzero,? is indefinite...............
Minimum rank Let G have n vertices. It is easy to obtain a matrix B S(G) with rank(b) = n (translate) It is easy to have full rank (use large diagonal) If G is the disjoint union of graphs G i then mr(g) = mr(g i ) Only connected graphs are studied If G is connected, mr(g) = 0 iff G is single vertex mr(g) = iff G = K n, n 2 mr(g) = n if and only if G is a path [Fiedler 69] Matrix and
Problem for Let T be a tree. (T ) is the maximum of p q such that there is a set of q vertices whose deletion leaves p paths. Theorem (Johnson, Leal Duarte 99) T mr(t ) = M(T ) = (T ) A related method for computing mr(t ) directly appeared earlier in [Nylen 96]. Numerous algorithms compute (T ) by using high degree ( ) vertices. The following algorithm works from the outside in. v is an outer high degree vertex if at most one component of T v contains high degree vertices. Matrix and Delete each outer high degree vertex. Repeat as needed.
Example Compute mr(t ) by computing (T ) = M(T ). Matrix and 2 4 5 6 8 7
Example Compute mr(t ) by computing (T ) = M(T ). Matrix and 2 4 5 6 8 7
Example Compute mr(t ) by computing (T ) = M(T ). Matrix and 4 2 5 6 7
Example Compute mr(t ) by computing (T ) = M(T ). Matrix and 2 5 6 7
Example Compute mr(t ) by computing (T ) = M(T ): the six vertices {, 2,, 5, 6, 7} were deleted there are 8 paths M(T ) = (T ) = 8 6 = 2 mr(t ) = 5 2 = 2 2 5 6 Matrix and 7
Adjacency matrix A(G) = A = [a ij ] where a ij = σ(a) = (α,..., α n ) Example W 5 2 5 4 { if i j 0 if i j 0 0 0 A = 0 0 0 0 0 0 (α, α 2, α, α 4, α 5 ) = ( 2, 5, 0, 0, + 5) Matrix and
If two graphs have different spectra (equivalently, different characteristic polynomials) of the adjacency matrix, then they are not isomorphic However, non-isomorphic graphs can be cospectral Example p(x) = x 6 7x 4 4x + 7x 2 + 4x Spectrally determined graphs: Complete graphs Graphs with one edge Empty graphs Graphs missing only edge Regular of degree 2 Regular of degree n- Matrix and K n,n,...,n mk n
Laplacian matrix L(G) = L = D A(G) where D = diag(deg(),..., deg(n)). σ(l) = (λ,..., λ n ) Example W 5 2 5 4 (λ, λ 2, λ, λ 4, λ 5 ) = (0,,, 5, 5) 4 0 L = 0 0 0 Matrix and
isomorphic graphs must have the same Laplacian spectrum (i.e., Laplacian characteristic polynomial) non-isomorphic graphs can be Laplacian cospectral [Schwenk 7], [McKay 77] For almost all trees T there is a non-somorphic tree T that has both the same adjacency spectrum and the same Lapalcian spectrum for any G, λ (G) = 0 algebraic connectivity: λ 2 (G), second smallest eigenvalue of L vertex connectivity: κ v (G), minimum number of vertices in a cutset (G K n ) [Fiedler 7] λ 2 (G) κ v (G) Matrix and Example λ 2 (W 5 ) = = κ v (W 5 )
Signless Laplacian matrix L (G) = L = D + A(G) where D = diag(deg(),..., deg(n)) σ( L ) = (µ,..., µ n ) Example W 5 2 5 4 4 0 L = 0 0 0 Matrix and (µ, µ 2, µ, µ 4, µ 5 ) = (, 9 7 2,,, 9+ 7 2 )
Normalized adjacency matrix  = D A D where D = diag(deg(),..., deg(n)) σ(â) = ( α,..., α n ) Example W 5 2 5 4  = ( α, α 2, α, α 4, α 5 ) = ( 2,, 0, 0, ) 0 2 2 2 2 2 0 0 2 0 0 2 0 0 2 0 0 Matrix and
Normalized Laplacian matrix L = D L D σ( L) = ( λ,..., λ n ) Example W 5 L = 2 5 2 2 2 2 2 4 2 2 2 0 0 0 0 ( λ, λ 2, λ, λ 4, λ 5 ) = (0,,, 4, 5 ) Matrix and
Normalized Signless Laplacian matrix L = D L D σ( L ) = ( µ,..., µ n ) Example W 5 L = 2 5 4 2 2 2 2 2 0 2 0 2 0 2 0 ( µ, µ 2, µ, µ 4, µ 5 ) = (, 2,,, 2) Matrix and
between A, L, L, Â, L, L and their spectra Let G be a graph of order n. Â + L = I So α n k+ + λ k = L + L = 2I So µ n k+ + λ k = 2 Example W 5 eigenvalue relationships ( α 5, α 4, α, α 2, α ) = (, 0, 0,, 2 ) ( λ, λ 2, λ, λ 4, λ 5 ) = (0,,, 4, 5 ) ( µ 5, µ 4, µ, µ 2, µ ) = (2,,, 2, ) Matrix and
Matrix and Let G be an r-regular graph of order n. L + A = ri, so λ k = r α n k+ L A = ri, so µ k = r + α k  = r A, so α k = r α k L = r L, so λ k = r λ k L = r L, so µ k = r µ k
A, L, L, Â, L, L and the incidence matrix Let G be a graph with n vertices and m edges. incidence matrix P = P(G) is the n m 0,-matrix with rows indexed by the vertices of G and columns indexed by the edges of{ G if e is incident with v P = [p ve ] where p ve = 0 otherwise L = PP T L = D L D = ( D P)( D P) T L = P P T where P is an oriented incidence matrix. L = D L D = ( D P )( D P ) T L, L, L, L are all positive semidefinite all eigenvalues λ k, µ k, λ k, µ k are nonnegative. Matrix and
Matrix and The spectral radius of B is ρ(b) = max β σ(b) β A, L, Â, L 0 (nonnegative) Theorem (Perron-Forbenius) Let P 0 be irreducible. Then ρ(p) > 0, ρ(p) is an eigenvalue of P, eigenvalue ρ(p) has a positive eigenvector, and ρ(p) is a simple eigenvalue of P (multiplicity ).
Matrix and Let G K n be connected. σ( L ) [0, 2] and µ n = 2 = ρ( L ) σ(â) [, ] and α n = = ρ(â) σ( L) [0, 2] and λ = 0 0 < λ 2 2 n n λ n 2 and λ n = 2 if and only if G is bipartite n λ i= i = n
Matrix and Colin de Verdière s graph parameters Colin de Verdière defined new graph parameters µ(g) and ν(g) minor monotone bound M from below use the Strong Arnold Property Unlike the specific matrices originally used in spectral graph theory, these parameters involve families of matrices Close connections with and minimum rank
A minor of G is a graph obtained from G by a sequence of edge deletions, vertex deletions, and edge contractions. A graph parameter ζ is minor monotone if for every minor H of G, ζ(h) ζ(g). X fully annihilates B if. BX = 0 2. X has 0 where B has nonzero. all diagonal elements of X are 0 The matrix B has the Strong Arnold Property (SAP) if the zero matrix is the only symmetric matrix that fully annihilates B. Matrix and
Matrix and See [van der Holst, Lovász, Shrijver 99] for information about SAP SAP comes from manifold theory. R B = {C : rank C = rank B}. S B = S(G(B)). B has SAP if and only if manifolds R B and S B intersect transversally at B. Transversal intersection allows perturbation.
µ(g) = max{null(l)} such that. L is a generalized Laplacian matrix (L S(G) and off-diagonal entries 0) 2. L has exactly one negative eigenvalue (with multiplicity one). L has SAP Theorem (Colin de Verdière 90) µ is minor monotone µ(g) if and only if G is a path µ(g) 2 if and only if G is outerplanar µ(g) if and only if G is planar Matrix and
Matrix and For any graph, µ(g) M(G). If G is not planar then < µ(g) M(G) is sometimes useful for small graphs To study minimum rank, generalized Laplacians and number of negative eigenvalues are not usually relevant
Matrix and Example 6 5 K 2,2,2 is planar but not outer planar, so µ(k 2,2,2 ) = (no generalized Laplacian of K 2,2,2 has rank 2) mr(k 2,2,2 ) = 2 and M(K 2,2,2 ) = 4 K 2,2,2 0 0 0 0 2 rank B = 2 0 0 0 = 2 0 0 4 0 2
Matrix and ν(g) = max{null(b)} such that. B S(G) 2. B is positive semi-definite. B has SAP [Colin de Verdière] ν is minor monotone For any graph, ν(g) M(G)
To study minimum rank, positive semi-definite is not usually relevant. Example No positive semi-definite matrix in S(K 2, ) has rank 2 so ν(k 2, ) = 2 mr(k 2, ) = 2 and M(K 2, ) = K 2, 0 0 0 0 rank B = 4 5 0 0 0 0 0 0 = 2 2 0 0 0 Matrix and
The new parameter ξ To study minimum rank, positive semi-definite, generalized Laplacians and number of negative eigenvalues usually are not relevant. Minor monotonicity is useful. Definition = max{null(b) : B S(G), B has SAP} Example: ξ(k 2,2,2 ) = 4 = M(K 2,2,2 ) because the matrix B has SAP Example: ξ(k 2, ) = = M(K 2, ) because the matrix B has SAP Matrix and
Matrix and For any graph G, µ(g) ν(g) M(G) ξ(p n ) = = M(P n ) ξ(k n ) = n = M(K n ) If T is a non-path tree, ξ(t ) = 2.
Matrix and Theorem (Barioli, Fallat, Hogben 05) ξ is minor monotone. Since ξ is minor monotone, the graphs G such that k can be characterized by a finite set of forbidden minors. if and only if G contains no K or K, minor. K K,
Theorem (Hogben, van der Holst 07) 2 if and only if G contains no minor in the T family. Matrix and K 4 K 2, T family T
Minimum rank problem Minimum rank is characterized for: trees [Nylen 96], [Johnson, Leal-Duarte 99] unicyclic graphs [Barioli, Fallat, Hogben 05] extreme minimum rank: mr(g) = 0,, 2: [Barrett, van der Holst, Loewy 04] mr(g) = G, G 2: [Fiedler 69], [Hogben, van der Holst 07], [Johnson, Loewy, Smith] Reduction techniques for cut-set of order [Barioli, Fallat, Hogben 04] and order 2 [van der Holst 08] joins [Barioli, Fallat 06] Matrix and
Minimum rank graph catalogs Minimum rank of many families of graphs determined at the 06 AIM Workshop. On-line catalogs of minimum rank for small graphs and families developed. The ISU group determined the order of all graphs of order 7. Minimum rank of families of graphs http://aimath.org/pastworkshops/catalog2.html Minimum rank of small of graphs http://aimath.org/pastworkshops/catalog.html Matrix and
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