Fractional Zero Forcing via Three-color Forcing Games

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.Comment: 24 page

Applications of analysis to the determination of the minimum number of distinct eigenvalues of a graph

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues of several families of graphs and small graphs

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at least the number of vertices of G less one are characterized.Comment: 26 pages; corrected statement of Theorem 3.5 (a

The copositive completion problem: Unspecified diagonal entries

In [L. Hogben, C.R. Johnson, R. Reams, The copositive matrix completion problem, Linear Algebra Appl. 408 (2005) 207–211] it was shown that any partial (strictly) copositive matrix all of whose diagonal entries are specified can be completed to a (strictly) copositive matrix. In this note we show that every partial strictly copositive matrix (possibly with unspecified diagonal entries) can be completed to a strictly copositive matrix, but there is an example of a partial copositive matrix with an unspecified diagonal entry that cannot be completed to a copositive matrix

Proof of a conjecture of Graham and Lov\'asz concerning unimodality of coefficients of the distance characteristic polynomial of a tree

We establish a conjecture of Graham and Lov\'asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal; we also prove they are log-concave

Spectral graph theory and the inverse eigenvalue problem of a graph

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with negative off-diagonal entries in the positions described by the edges of the graph ( and zero in every other off-diagonal position).The set of all real symmetric matrices having nonzero off-diagonal entries exactly where the graph G has edges is denoted by S( G). Given a graph G, the problem of characterizing the possible spectra of B, such that B. S( G), has been referred to as the Inverse Eigenvalue Problem of a Graph. In the last fifteen years a number of papers on this problem have appeared, primarily concerning trees.The adjacency matrix and Laplacian matrix of G and their normalized forms are all in S( G). Recent work on generalized Laplacians and Colin de Verdiere matrices is bringing the two areas closer together. This paper surveys results in Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph, examines the connections between these problems, and presents some new results on construction of a matrix of minimum rank for a given graph having a special form such as a 0,1-matrix or a generalized Laplacian

Matrix completion problems for pairs of related classes of matrices

For a class X of real matrices, a list of positions in an n×n matrix (a pattern) is said to have X-completion if every partial X-matrix that specifies exactly these positions can be completed to an X-matrix. If X and X0 are classes that satisfy the conditions any partial X-matrix is a partial X0-matrix, for any X0-matrix A and ε\u3e0, A+εI is a X-matrix, and for any partial X-matrix A, there exists δ\u3e0 such that A−δĨ is a partial X-matrix (where Ĩ is the partial identity matrix specifying the same pattern as A) then any pattern that has X0-completion must also have X-completion. However, there are usually patterns that have X-completion that fail to have X0-completion. This result applies to many pairs of subclasses of P- and P0-matrices defined by the same restriction on entries, including the classes P/P0-matrices, (weakly) sign-symmetric P/P0-matrices, and non-negative P/P0-matrices. It also applies to other related pairs of subclasses of P0-matrices, such as the pairs classes of P/P0,1-matrices, (weakly) sign-symmetric P/P0,1-matrices and non-negative P/P0,1-matrices. Furthermore, any pattern that has (weakly sign-symmetric, sign-symmetric, non-negative) P0-completion must also have (weakly sign-symmetric, sign-symmetric, non-negative) P0,1-completion, although these pairs of classes do not satisfy condition (3). Similarly, the class of inverse M-matrices and its topological closure do not satisfy condition (3), but the conclusion remains true, and the matrix completion problem for the topological closure of the class of inverse M-matrices is solved for patterns containing the diagonal