Item Open AccessThe enhanced principal rank characteristic sequence for skew-symmetric matrices(Elsevier, 2015-08-15) Fallat, Shaun; Olesky, Dale; van den Driessche, PaulineThe enhanced principal rank characteristic sequence (epr-sequence) was originally defined for an n ×n real symmetric matrix or an n ×n Hermitian matrix. Such a sequence is defined to be l1l2···ln where lk is A,S, or N depending on whether all, some, or none of the matrix principal minors of order k are nonzero. Here we give a complete characterization of the attainable epr-sequences for real skew-symmetric matrices. With the constraint that lk=0 if k is odd, we show that nearly all epr-sequences are attainable by skew-symmetric matrices, which is in contrast to the case of real symmetric or Hermitian matrices for which many epr-sequences are forbidden. ©2015 Item Open AccessOn the complexity of the positive semidefinite zero forcing number(Elsevier, 2015) Meagher, Karen; Fallat, Shaun; Yang, BotingThe positive semidefinite zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring and is related to a more conventional version of zero forcing. We establish a relation between the zero forcing and the fast–mixed searching, which implies some NP-completeness results for the zero forcing problem. Relationships between positive semidefinite zero forcing sets and clique coverings are well-understood for chordal graphs. Building upon constructions associated with optimal tree covers and forest covers, we present a linear time algorithm for computing the positive semidefinite zero forcing number of chordal graphs. We also prove that it is NP-complete to determine whether a graph has a positive semidefinite zero forcing set with an additional property. Item Open AccessParameters Related to Tree-Width, Zero Forcing, and Maximum Nullity of a Graph(Wiley Periodicals, Inc., 2013) Barioli, Francesco; Barrett, Wayne; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; Shader, Bryan; van den Driessche, Pauline; van der Holst, HeinTree-width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdi`ere type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree-width, largeur d’arborescence, path-width, and proper path-width are each characterized in terms of a minor monotone floor of a certain zero forcing parameter defined by a color change rule. Item Open AccessOn the relationship between zero forcing number and certain graph coverings(2014) Alinaghipour, Fatemeh; Fallat, Shaun; Meagher, KarenThe zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block- cycle graph the zero forcing number equals the path cover number. We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex. Item Open AccessThe principal rank characteristic sequence over various fields(Elsevier, 2014) Barrett, Wayne; Butler, Steve; Catral, Minnie; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; van den Driessche, Pauline; Young, MichaelGiven an n-by-n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0; 1,..., n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such attainable sequences determined for all n over any field with characteristic 2. A complete list of attainable sequences for real symmetric matrices of order 7 is reported. Item Open AccessThe enhanced principal rank characteristic sequence(Elsevier, 2015) Butler, Steve; Catral, Minnie; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; van den Driessche, Pauline; Young, michaelThe enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n ×n matrix is a sequence from A,S, or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the kth entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero). Various techniques including the Schur complement are introduced to establish that certain subsequences such as NAN are forbidden in epr-sequences over fields of characteristic not two. Using probabilistic methods over fields of characteristic zero, it is shown that any sequence ofAs andSs ending inAis attainable, and any sequence ofAs andSs followed by one or moreNs is attainable; additional families of attainable epr-sequences are constructed explicitly by other methods. For real symmetric matrices of orders 2, 3, 4, and 5, all attainable epr-sequences are listed with justifications. Item Open AccessOn the null space struture associted with trees and cycles(The Charles Babbage Research Centre, 2013-02) Fallat, Shaun; Nasserasr, ShahlaIn this work, we study the structure of the null spaces of matrices associated with graphs. Our primary tool is utilizing Schur complements based on certain collections of independent vertices. This idea is applied in the case of trees, and seems to represent a unifying theory within the context of the support of the null space. We extend this idea and apply it to describe the null vectors and corresponding nullities of certain symmetric matrices associated with cycles. Item Open AccessColin de Verdiere parameters of chordal graphs(International Linear Algebra Society, 2013-01) Mitchell, Lon; Fallat, ShaunThe Colin de Verdi`ere parameters mu and nu are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For nu the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on computing μ for split graphs. Item Open AccessMinimum number of distinct eigenvalues of graphs(International Linear Algebra Society, 2013-09) Ahmadi, Bahman; Alinaghipour, Fatemeh; Cavers, Michael; Fallat, Shaun; Meagher, Karen; Nasserasr, ShahlaThe minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity. Item Open AccessNote on Nordhaus-Gaddum problems for Colin de Verdiere type parameters(Public Knowledge Network, 2013) Barrett, Wayne; Fallat, Shaun; Hall, Tracy; Hogben, LeslieWe establish the bounds 4 on the Nordhaus- Gaddum sum upper bound multipliers for all graphs G, in connections with certain Colin de Verdi ere type graph parameters. The Nordhaus-Gaddum sum lower bound is conjectured to be |G|-2, and if these parameters are replaced by the maximum nullity M(G), this bound is called the Graph Complement Conjecture in the study of minimum rank/maximum nullity problems. Item Open AccessThe maximum nullity of a complete edge subdivision graph is equal to it zero forcing number(International Linear Algebra Society, 2014-06) Barrett, Wayne; Butler, Steve; Catral, Minnie; Hall, Tracy; Fallat, Shaun; Hogben, Leslie; Young, MichaelBarrett et al. asked in [W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530–563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F, °G) = Z(°G) by introducing the bridge tree of a connected graph. Since this equality is valid for all fields, °G has field independent minimum rank, and we also show that °G has a universally optimal matrix.