Geršgorin discs revisited

Fiedler, Miroslav; Hall, Frank J.; Marsli, Rachid

doi:10.1016/j.laa.2012.07.027

Cited by 6 publications

(5 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Varga's nice book [31] surveys various applications and extensions of this important theorem. Recently, Marsli and Hall [25] found an interesting result, which states that if λ is an eigenvalue of an n × n matrix A with geometric multiplicity k, then λ is in at least k of the n Gerŝgorin discs of A. Fiedler et al [12] proved that for a triple of positive integers k, r, t with k ≤ r ≤ t, there is a t × t complex matrix A and an eigenvalue λ of A such that λ has geometric multiplicity k and algebraic multiplicity t, and λ is in precisely r Gerŝgorin discs of A. Marsli and Hall extended these results in subsequent papers [24,26,27]. Bárány and Solymosi [6] showed that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk.…”

mentioning

confidence: 99%

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

Atik

Panigrahi

2018

ELA

View full text Add to dashboard Cite

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

show abstract

mentioning

confidence: 99%

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

Atik

Panigrahi

2018

ELA

View full text Add to dashboard Cite

show abstract

“…An interesting and recent theorem of Marsli and Hall [5] states that if an eigenvalue of a matrix A has geometric multiplicity k, then it lies in at least k of the Gershgorin disks of A. They have extended this result in subsequent papers [3,6,7,8]. Here we focus on the k = 2 case for non-negative matrices.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

Bárány

Solymosi

2017

A Journey Through Discrete Mathematics

View full text Add to dashboard Cite

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper. 1 arXiv:1609.07439v1 [math.CO]

show abstract

“…A matrix A ∈ M n has n Geršgorin discs D i , some of which may degenerate into points and some of which may be duplicates, as in the trivial example of an identity matrix. Recently, the authors extended the Geršgorin theory in the articles [1], [3], [4], and [5]. In particular, the following result was proved in [3].…”

Section: Geršgorin Theorem Let a ∈ M N And Letmentioning

confidence: 99%