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$(0, 1)$ -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

Let $m$ and $n$ be positive integers, and let $R = (r_{1}, ..., r_{m})$ and $S = (s_{1}, ..., s_{n})$ be nonnegative integral vectors. Let $A (R, S)$ be the set of all $m \times n$ $(0, 1)$ -matrices with row sum vector $R$ and column vector...

$𝒟_{n, r}$ is not potentially nilpotent for $n \geq 4 r - 2$

Yan Ling Shao, Yubin Gao, Wei Gao (2016)

Czechoslovak Mathematical Journal

An $n \times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$ . Let $𝒟_{n, r}$ be an $n \times n$ sign pattern with $2 \leq r \leq n$ such that the superdiagonal and the $(n, n)$ entries are positive, the $(i, 1)$ $(i = 1 ...$

A characterization of singular graphs.

Sciriha, Irene (2007) ELA. The Electronic Journal of Linear Algebra [electronic only]

A combinatorial approach to the conditioning of a single entry in the stationary distribution for a Markov chain.

Kirkland, S. (2004) ELA. The Electronic Journal of Linear Algebra [electronic only]

A Fiedler-like theory for the perturbed Laplacian

Israel Rocha, Vilmar Trevisan (2016) Czechoslovak Mathematical Journal

The perturbed Laplacian matrix of a graph

G

is defined as

L^{D} = D - A

, where

D

is any diagonal matrix and

A

is a weighted adjacency matrix of

G

. We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore, we use...

A formula for all minors of the adjacency matrix and an application

R. B. Bapat, A. K. Lal, S. Pati (2014)

Special Matrices

We supply a combinatorial description of any minor of the adjacency matrix of a graph. This description is then used to give a formula for the determinant and inverse of the adjacency matrix, A(G), of a graph G, whenever A(G) is invertible, where G is formed by replacing the edges of a tree by path bundles.