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𝒟 n , r is not potentially nilpotent for n 4 r - 2

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

Czechoslovak Mathematical Journal

An n × 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 × n sign pattern with 2 r n such that the superdiagonal and the ( n , n ) entries are positive, the ( i , 1 ) ( i = 1 ...

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.

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