The Experts below are selected from a list of 5121 Experts worldwide ranked by ideXlab platform
Leslie Hogben - One of the best experts on this subject based on the ideXlab platform.
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zero forcing and maximum Nullity for hypergraphs
Discrete Applied Mathematics, 2019Co-Authors: Leslie HogbenAbstract:Abstract The concept of zero forcing is extended from graphs to uniform hypergraphs in analogy with the way zero forcing was defined as an upper bound for the maximum Nullity of the family of symmetric matrices whose nonzero pattern of entries is described by a given graph: A family of symmetric hypermatrices is associated with a uniform hypergraph and zeros are forced in a null vector. The value of the hypergraph zero forcing number and maximum Nullity are determined for various families of uniform hypergraphs and the effects of several graph operations on the hypergraph zero forcing number are determined. The hypergraph zero forcing number is compared to the infection number of a hypergraph and the iteration process in hypergraph power domination.
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minimum rank maximum Nullity and zero forcing number of simple digraphs
Electronic Journal of Linear Algebra, 2013Co-Authors: Adam H Berliner, Leslie Hogben, My Huynh, Minerva Catral, Kelsey Lied, Michael YoungAbstract:A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum Nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum Nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simple digraphs are established. The effect of deletion of a vertex on minimum rank or zero forcing number is analyzed, and simple digraphs having very low or very high zero forcing number are characterized.
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vertex and edge spread of zero forcing number maximum Nullity and minimum rank of a graph
Linear Algebra and its Applications, 2012Co-Authors: Christina Edholm, Leslie Hogben, My Huynh, Joshua LagrangeAbstract:Abstract The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ≠ j ) is nonzero whenever { i , j } is an edge in G and is zero otherwise; maximum Nullity is taken over the same set of matrices. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum Nullity from above. The spread of a graph parameter at a vertex v or edge e of G is the difference between the value of the parameter on G and on G - v or G - e . Rank spread (at a vertex) was introduced in [4] . This paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank/maximum Nullity and zero forcing number. Properties of the spreads are established and used to determine values of the minimum rank/maximum Nullity and zero forcing number for various types of grids with a vertex or edge deleted.
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note on positive semidefinite maximum Nullity and positive semidefinite zero forcing number of partial 2 trees
Electronic Journal of Linear Algebra, 2012Co-Authors: Jason Ekstrand, Leslie Hogben, Craig Erickson, J RoatAbstract:The maximum positive semidefinite Nullity of a multigraph G is the largest possible Nullity over all real positive semidefinite matrices whose (i,j)th entry (for i 6 j) is zero if i and j are not adjacent in G, is nonzero if fi,jg is a single edge, and is any real number if fi,jg is a multiple edge. The definition of the positive semidefinite zero forcing number for simple graphs is extended to multigraphs; as for simple graphs, this parameter bounds the maximum positive semidefinite Nullity from above. The tree cover number T(G) is the minimum number of vertex disjoint induced simple trees that cover all of the vertices of G. The result that M+(G) = T(G) for an outerplanar multigraph G (F. Barioli et al. Minimum semidefinite rank of outerplanar graphs and the tree cover number. Electron. J. Linear Algebra, 22:10-21, 2011.) is extended to show that Z+(G) = M+(G) = T(G) for a multigraph G of tree-width at most 2.
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a note on minimum rank and maximum Nullity of sign patterns
Electronic Journal of Linear Algebra, 2011Co-Authors: Leslie HogbenAbstract:The minimum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. The maximum Nullity of a sign pattern is the largest possible Nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph parameters that have been used to bound maximum Nullity of a zero-nonzero pattern, including path cover number and edit distance, are extended to sign patterns, and the SNS number is introduced to usefully generalize the triangle number to sign patterns. It is shown that for tree sign patterns (that need not be combinatorially symmetric), minimum rank is equal to SNS number, and maximum Nullity, path cover number and edit distance are equal, providing a method to compute minimum rank for tree sign patterns. The minimum rank of small sign patterns is determined.
Camila A Ramirez - One of the best experts on this subject based on the ideXlab platform.
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minimum rank maximum Nullity and zero forcing number for selected graph families
Involve A Journal of Mathematics, 2010Co-Authors: Edgard Almodovar, Leslie Hogben, Laura Deloss, Kirsten Hogenson, Kaitlyn Murphy, Travis Peters, Camila A RamirezAbstract:The minimum rank of a simple graph G is dened to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i 6 j) is nonzero whenever fi;jg is an edge in G and is zero otherwise. Maximum Nullity is taken over the same set of matrices, and the sum of maximum Nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum Nullity from above. This paper denes the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide the examples showing that the maximum Nullity of a graph and its dual may dier, and similarly for zero forcing number.
Travis Peters - One of the best experts on this subject based on the ideXlab platform.
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positive semidefinite maximum Nullity and zero forcing number
Electronic Journal of Linear Algebra, 2012Co-Authors: Travis PetersAbstract:The zero forcing number Z(G) is used to study the minimum rank/maximum Nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidef- inite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum Nullity and zero forcing number for a variety of graph families are computed. In addition, field independence of the minimum rank of the hypercube is established, by showing there is a positive semidefinite matrix that is universally optimal.
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minimum rank maximum Nullity and zero forcing number for selected graph families
Involve A Journal of Mathematics, 2010Co-Authors: Edgard Almodovar, Leslie Hogben, Laura Deloss, Kirsten Hogenson, Kaitlyn Murphy, Travis Peters, Camila A RamirezAbstract:The minimum rank of a simple graph G is dened to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i 6 j) is nonzero whenever fi;jg is an edge in G and is zero otherwise. Maximum Nullity is taken over the same set of matrices, and the sum of maximum Nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum Nullity from above. This paper denes the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide the examples showing that the maximum Nullity of a graph and its dual may dier, and similarly for zero forcing number.
Edgard Almodovar - One of the best experts on this subject based on the ideXlab platform.
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minimum rank maximum Nullity and zero forcing number for selected graph families
Involve A Journal of Mathematics, 2010Co-Authors: Edgard Almodovar, Leslie Hogben, Laura Deloss, Kirsten Hogenson, Kaitlyn Murphy, Travis Peters, Camila A RamirezAbstract:The minimum rank of a simple graph G is dened to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i 6 j) is nonzero whenever fi;jg is an edge in G and is zero otherwise. Maximum Nullity is taken over the same set of matrices, and the sum of maximum Nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum Nullity from above. This paper denes the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide the examples showing that the maximum Nullity of a graph and its dual may dier, and similarly for zero forcing number.
Tingzeng Wu - One of the best experts on this subject based on the ideXlab platform.
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on the permanental Nullity and matching number of graphs
Linear & Multilinear Algebra, 2018Co-Authors: Tingzeng WuAbstract:For a graph G with n vertices, let and A(G) denote the matching number and adjacency matrix of G, respectively. The permanental polynomial of G is defined as . The permanental Nullity of G, denoted by , is the multiplicity of the zero root of . In this paper, we use the Gallai–Edmonds structure theorem to derive a concise formula which reveals the relationship between the permanental Nullity and the matching number of a graph. Furthermore, we prove a necessary and sufficient condition for a graph G to have . As applications, we show that every unicyclic graph G on n vertices satisfies , that the permanental Nullity of the line graph of a graph is either zero or one and that the permanental Nullity of a factor critical graph is always zero.
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on the permanental Nullity and matching number of graphs
arXiv: Combinatorics, 2016Co-Authors: Tingzeng WuAbstract:For a graph $G$ with $n$ vertices, let $\nu(G)$ and $A(G)$ denote the matching number and adjacency matrix of $G$, respectively. The permanental polynomial of $G$ is defined as $\pi(G,x)={\rm per}(Ix-A(G))$. The permanental Nullity of $G$, denoted by $\eta_{per}(G)$, is the multiplicity of the zero root of $\pi(G,x)$. In this paper, we use the Gallai-Edmonds structure theorem to derive a concise formula which reveals the relationship between the permanental Nullity and the matching number of a graph. Furthermore, we prove a necessary and sufficient condition for a graph $G$ to have $\eta_{per}(G)=0$. As applications, we show that every unicyclic graph $G$ on $n$ vertices satisfies $n-2\nu(G)-1 \le \eta_{per}(G) \le n-2\nu(G)$, that the permanental Nullity of the line graph of a graph is either zero or one, and that the permanental Nullity of a factor critical graph is always zero.
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per spectral characterizations of graphs with extremal per Nullity
Linear Algebra and its Applications, 2015Co-Authors: Tingzeng Wu, Heping ZhangAbstract:Abstract A graph G is said to be determined by its permanental spectrum if any graph having the same permanental spectrum as G is isomorphic to G . In this paper, we introduce the permanental Nullity of a graph, the multiplicity of the number zero in the permanental spectrum of a graph, to study graphs determined by their permanental spectra. First, we determine all graphs of order n whose permanental nullities are n − 2 , n − 3 , n − 4 and n − 5 , respectively. Then, we show that all graphs with the permanental Nullity n − 2 , n − 3 , or n − 5 , and all non-bipartite graphs with the permanental Nullity n − 4 are determined by their permanental spectra. In particular, we prove that the complete bipartite graphs are determined by their permanental spectra.
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ICICA (2) - On the Nullity Algorithm of Tree and Unicyclic Graph
Communications in Computer and Information Science, 2010Co-Authors: Tingzeng Wu, Defu MaAbstract:Let G be a graph with n vertices and q(G) be the maximum matching number of G. Let η(G) denote the Nullity of G (the multiplicity of the eigenvalue zero of G). It is shown that the Nullity algorithm of tree and unicyclic graph. At the same time, to prove two algorithms are efficient.
