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The scrambling index of an $n \times n$ primitive Boolean matrix $A$ is the smallest positive integer $k$ such that $A^{k} {(A^{T})}^{k} = J$ , where $A^{T}$ denotes the transpose of $A$ and $J$ denotes the $n \times n$ all ones matrix. For an $m \times n$ Boolean matrix $M$ , its Boolean rank $b (M)$ is the smallest positive integer $b$ such that $M = A B$ for some $m \times b$ Boolean matrix $A$ and $b \times n$ Boolean matrix $B$ . In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an $n \times n$ primitive matrix $M$ in terms of its Boolean rank $b (M)$ , and they also characterized all primitive...

The $Q$ -matrix completion problem.

Dealba, Luz Maria, Hogben, Leslie, Sarma, Bhaba Kumar (2009)

ELA. The Electronic Journal of Linear Algebra [electronic only]

The Randić energy of generalized double sun

Wei Gao (2022)

Czechoslovak Mathematical Journal

We show that the family of trees defined as generalized double sun of odd order satisfies the conjecture for the Randić energy proposed by I. Gutman, B. Furtula, S. B. Bozkurt (2014).

The rank of a cograph.

Royle, Gordon F. (2003)

The Electronic Journal of Combinatorics [electronic only]

The real symmetric matrices of odd order with a P-set of maximum size

Zhibin Du, Carlos M. da Fonseca (2016)

Czechoslovak Mathematical Journal

Suppose that $A$ is a real symmetric matrix of order $n$ . Denote by $m_{A} (0)$ the nullity of $A$ . For a nonempty subset $α$ of ${1, 2, ..., n}$ , let $A (α)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $α$ . When $m_{A (α)} (0) = m_{A} (0) + | α |$ , we call $α$ a P-set of $A$ . It is known that every P-set of $A$ contains at most $⌊ n / 2 ⌋$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step...

The scattering matrix of a graph.

Mizuno, Hirobumi, Sato, Iwao (2008)

The Electronic Journal of Combinatorics [electronic only]

The signless Laplacian separator of graphs.

You, Zhifu, Liu, Bolian (2011)

ELA. The Electronic Journal of Linear Algebra [electronic only]

The signless Laplacian spectral radius of bicyclic graphs with a given girth.

Zhu, Zhongxun (2011)

ELA. The Electronic Journal of Linear Algebra [electronic only]

The signless Laplacian spectral radius of graphs with given number of cut vertices

Lin Cui, Yi-Zheng Fan (2010)

Discussiones Mathematicae Graph Theory

In this paper, we determine the graph with maximal signless Laplacian spectral radius among all connected graphs with fixed order and given number of cut vertices.

The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with $n$ vertices and $k$ pendant vertices

Muhuo Liu, Xuezhong Tan, Bo Lian Liu (2010)

Czechoslovak Mathematical Journal

In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with $n$ vertices and $k$ pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with $n$ vertices and $k$ pendant vertices, respectively....

The singularity/nonsingularity problem for matrices satisfying diagonal dominance conditions formulated in terms of directed graphs.

Kolotilina, L.Yu. (2004)

Zapiski Nauchnykh Seminarov POMI

The spectral determinations of the connected multicone graphs $K_{w} ▽ m P_{17}$ and $K_{w} ▽ m S$

Ali Zeydi Abdian, S. Morteza Mirafzal (2018)

Czechoslovak Mathematical Journal

Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let $K_{w}$ denote a complete graph on $w$ vertices, and let $m$ be a positive integer number. In A. Z. Abdian (2016) it has been...

The spectral excess theorem for distance-regular graphs: a global (over)view.

van Dam, Edwin R. (2008)

The Electronic Journal of Combinatorics [electronic only]

The spectral gap of random graphs with given expected degrees.

Coja-Oghlan, Amin, Lanka, André (2009)

The Electronic Journal of Combinatorics [electronic only]

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