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Three-connected graphs whose maximum nullity is at most three

Linear Algebra and its Applications, 2008
Let \(G=(V,E)\) be a graph with \(V=\{1,2,\dots,n\}\). Define \(\mathcal S(G)\) as the set of all \(n\times n\) real-valued symmetric matrices \(A=[a_{i,j}]\) with \(a_{i,j}\neq 0\), \(i\neq j\), if and only if \(ij\in E\). The maximum nullity of \(G\), denoted by \(M(G)\), is the largest possible nullity of any matrix \(A\in\mathcal S(G)\).
Holst, van der, H.
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Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees [PDF]

The Electronic Journal of Linear Algebra, 2012
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.
Ekstrand, Jason, Erickson, Craig, Hay, Diana, Hogben, Leslie, Roat, Jolie   +4 more
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The maximum nullity of a complete subdivision graph is equal to its zero forcing number [PDF]

The Electronic Journal of Linear Algebra, 2014
Barrett 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.
Barrett, Wayne, Butler, Steve, Catral, Minerva, Fallat, Shaun, Hall, H. Tracy, Hogben, Leslie, Young, Michael   +6 more
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The stable maximum nullity of digraphs and $1$-DAGs

Given a digraph $D=(V,A)$ with vertex-set $V=\{1,\ldots,n\}$ and arc-set $A$, we denote by $Q(D)$ the set of all real $n\times n$ matrices $B=[b_{u,w}]$ with $b_{u,u}\not=0$ for all $u\in V$, $b_{u,w} \not= 0$ if $u\not=w$ and there is an arc from $u$ to $w$, and $b_{u,w}=0$ if $u\not=w$ and there is no arc from $u$ to $w$.
Arav, Marina, van der Holst, Hein
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The maximum nullity of a complete edge subdivision graph is equal to it zero forcing number

, 2014
Barrett 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
Barrett, Wayne, Butler, Steve, Catral, Minnie, Hall, Tracy, Fallat, Shaun, Hogben, Leslie, Young, Michael   +6 more
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Two-connected signed graphs with maximum nullity at most two [PDF]

Linear Algebra and its Applications, 2021
A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,\ldots,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd and the other edges of $E$ even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are ...
Marina Arav, Frank J. Hall, Zhongshan Li, Hein van der Holst   +3 more
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Zero forcing and maximum nullity for hypergraphs [PDF]

, 2021
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 ...
Hogben, Leslie
core  

Line graphs: their maximum nullities and zero forcing numbers [PDF]

, 2008
summary:The maximum nullity over a collection of matrices associated with a graph has been attracting the attention of numerous researchers for at least three decades.
Fallat, Shaun, Wang, Wendy, Johnson, Charles, Harris, Benjamin, Lay, Margaret, Soltani, Abolghasem, Mitchell, Lon, Narayan, Sivaram, Steinmetz, Kelly, Sutton, Brian, Hackney, Philip, Booth, Mathew, Pascoe, Amanda   +12 more
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Maximum nullity of Cayley graph

, 2017
One of the most interesting problems on maximum nullity (minimum rank) is to characterize $M(\mathcal{G})$ ($mr(\mathcal{G})$) for a graph $\mathcal{G}$. In this regard, many researchers have been trying to find an upper or lower bound for the maximum nullity. For more results on this topic, see \cite{4}, \cite{2}, \cite{10} and \cite{1}. In this paper,
Vatandoost, Ebrahim, Pour, Yasser Golkhandy   +1 more
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On the nullity of bicyclic graphs

, 2008
The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we obtain the nullity set of bicyclic graphs of order n, and determine the bicyclic graphs with maximum ...
Liu, Bolian, Hu, Shengbiao, Xuezhong, Tan   +2 more
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