Linear Algebra and its Applications最新文献

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Quadratic embedding constants of fan graphs and graph joins

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-06 DOI: 10.1016/j.laa.2025.01.001

Wojciech Młotkowski , Nobuaki Obata

We derive a general formula for the quadratic embedding constant of a graph join

{\bar{K}}_{m} + G

, where

{\bar{K}}_{m}

is the empty graph on

m \geq 1

vertices and G is an arbitrary graph. Applying our formula to a fan graph

K_{1} + P_{n}

, where

K_{1} = {\bar{K}}_{1}

is the singleton graph and

P_{n}

is the path on

n \geq 1

vertices, we show that

QEC (K_{1} + P_{n}) = - {\tilde{α}}_{n} - 2

, where

{\tilde{α}}_{n}

is the minimal zero of a new polynomial

Φ_{n} (x)

related to Chebyshev polynomials of the second kind. Moreover, for an even n we have

{\tilde{α}}_{n} = \min ev (A_{n})

, where the right-hand side is the minimal eigenvalue of the adjacency matrix

A_{n}

P_{n}

. For an odd n we show that

\min ev (A_{n + 1}) \leq {\tilde{α}}_{n} < \min ev (A_{n})

引用次数: 0

Threshold graphs, Kemeny's constant, and related random walk parameters

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-06 DOI: 10.1016/j.laa.2024.12.022

Jane Breen , Sooyeong Kim , Alexander Low Fung , Amy Mann , Andrei A. Parfeni , Giovanni Tedesco

Kemeny's constant measures how fast a random walker moves around in a graph. Expressions for Kemeny's constant can be quite involved, and for this reason, many lines of research focus on graphs with structure that makes them amenable to more in-depth study (for example, regular graphs, acyclic graphs, and 1-connected graphs). In this article, we study Kemeny's constant for random walks on threshold graphs, which are an interesting family of graphs with properties that make examining Kemeny's constant difficult; that is, they are usually not regular, not acyclic, and not 1-connected. This article is a showcase of various techniques for calculating Kemeny's constant and related random walk parameters for graphs. We establish explicit formulae for

K (G)

in terms of the construction code of a threshold graph, and completely determine the ordering of the accessibility indices of vertices in threshold graphs.

引用次数: 0

Symmetric doubly stochastic inverse eigenvalue problem for odd sizes

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-31 DOI: 10.1016/j.laa.2024.12.020

Mohadese Raeisi Sarkhoni , Hossein Momenaee Kermani , Azim Rivaz

The symmetric doubly stochastic inverse eigenvalue problem seeks to determine the necessary and sufficient conditions for a real list of eigenvalues to be realized by a symmetric doubly stochastic matrix. Nader et al. (2019) [15], established that for odd integers n a list of the form

σ = (1, λ_{2}, λ_{3}, . . ., λ_{_{n - 1}}, - 1)

with

| λ_{i} | < 1

for

i = 2, . . ., n - 1

cannot be the spectrum of any

n \times n

doubly stochastic matrix. This implies that the list

σ = (1, 0, . . ., 0, - 1)

is also unrealizable.

This paper extends these findings by proving that for odd n and

λ_{_{n}} \in [- 1, - \frac{n - 1}{n})

, the list

(1, 0, . . ., 0, λ_{_{n}})

cannot be the spectrum of a symmetric doubly stochastic matrix. We demonstrate that for odd n the list

σ = (1, 0, . . ., 0, - \frac{n - 1}{n})

, is indeed realizable as the spectrum of a symmetric doubly stochastic matrix.

Furthermore, we utilize our methodology to derive new sufficient conditions for the existence of

n \times n

symmetric doubly stochastic matrices with a prescribed list of eigenvalues. This leads to a condition for the existence of symmetric doubly stochastic matrices with a normalized Suleimanova spectrum. The paper concludes with additional partial results and illustrative examples.

引用次数: 0

Linear bijective maps preserving invertibility on pairs of similar matrices

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-31 DOI: 10.1016/j.laa.2024.12.021

Constantin Costara

Let

n \geq 2

be a natural number, and denote by

M_{n}

the space of all

n \times n

matrices over the complex field. In this paper, we characterize linear bijective maps φ on

M_{n}

having the property that if

A, B \in M_{n}

are similar matrices and

φ (A)

is invertible, then

φ (B)

is invertible as well.

引用次数: 0

Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with fractional Laplacian and variable coefficient wave number μ

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-27 DOI: 10.1016/j.laa.2024.12.015

Andrea Adriani , Rosita L. Sormani , Cristina Tablino-Possio , Rolf Krause , Stefano Serra-Capizzano

The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a fractional Laplacian and a variable coefficient wave number μ, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix sequences, with the main novelty regarding a complete picture of the case where

μ = μ (x, y)

is nonconstant. We report numerical evidence supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.

引用次数: 0

Some stability results for spectral extremal problems of graphs with bounded matching number

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-20 DOI: 10.1016/j.laa.2024.12.018

Shixia Jiang , Xiying Yuan , Yanni Zhai

For a set of graphs

H

, a graph is called

H

-free if it does not contain any member of

H

as a subgraph. The maximum value of spectral radius among all

H

-free graphs of order n is denoted by

spex (n, H)

, and the set of corresponding extremal graphs is denoted by

SPEX (n, H)

. In this paper, we give a stability result for graphs in

SPEX (n, H)

when

spex (n, H) \geq \sqrt{s (n - s)}

and

ex (n, H) \leq s n

. As an application, we may give some characterizations for the graphs in

SPEX (n, {M_{s + 1}, H})

, where

M_{s + 1}

is a matching with

s + 1

edges and H is any non-bipartite graph.

引用次数: 0

Remarks on generating families of matrix algebras of small orders

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-20 DOI: 10.1016/j.laa.2024.12.013

Yaroslav Shitov

Let

n ⩽ 7

, and let S be a family of

n \times n

matrices over a field

F

. I prove that the

F

-linear span of

{(S \cup {Id})}^{2 n - 2}

is the algebra generated by S.

引用次数: 0

Fractional perfect matching and distance spectral radius in graphs

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-19 DOI: 10.1016/j.laa.2024.12.016

Lei Zhang , Yaoping Hou , Haizhen Ren

A fractional matching of a graph G is a function f giving each edge a number in

[0, 1]

so that

\sum_{e \in E_{G} (v)} f (e) \leq 1

for each

v \in V (G)

, where

E_{G} (v)

is the set of edges incident to v. In this paper, we give a distance spectral radius condition to guarantee the existence of a fractional perfect matching. This result generalize the result of Lin and Zhang (2021) [22].

引用次数: 0

On the necessary and sufficient conditions for Hadamard-Fischer-Koteljanskii type inequalities

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-19 DOI: 10.1016/j.laa.2024.12.017

Phillip Braun , Hristo Sendov

This work explores the ratios of products of determinants of principal submatrices of positive definite matrices. We investigate conditions under which these ratios are bounded, particularly revisiting the necessary/sufficient conditions proposed by Johnson and Barrett. This analysis extends to set-theoretic consequences and unboundedness of certain ratios. We also demonstrate how these conditions can be used to prove the boundedness of several known determinantal inequalities. Additionally, we address the optimization problem of finding the supremum of such ratios over all positive definite matrices, formulating it as a linear optimization program. Finally, for completeness, we include the proofs of theorems that appear to have been previously known but lack accessible proofs.

引用次数: 0

On the local dimensions of solutions of Brent equations

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-19 DOI: 10.1016/j.laa.2024.12.011

Xin Li , Yixin Bao , Liping Zhang

Let

〈 m, n, p 〉

be the matrix multiplication tensor. The solution set of Brent equations corresponds to the tensor decompositions of

〈 m, n, p 〉

. We study the local dimensions of solutions of the Brent equations over the field of complex numbers. The rank of Jacobian matrix of Brent equations provides an upper bound of the local dimension, which is well-known. We calculate the ranks for some typical known solutions, which are provided in the databases [16] and [17]. We show that the automorphism group of the natural algorithm computing

〈 m, n, p 〉

(P_{m} \times P_{n} \times P_{p}) ⋊ Q (m, n, p)

, where

P_{m}

P_{n}

and

P_{p}

are groups of generalized permutation matrices,

Q (m, n, p)

is a subgroup of

S_{3}

depending on m, n and p. For other algorithms computing

〈 m, n, p 〉

, some conditions are given, which imply the corresponding automorphism groups are isomorphic to subgroups of

(P_{m} \times P_{n} \times P_{p}) ⋊ Q (m, n, p)

. So under these conditions,

m^{2} + n^{2} + p^{2} - m - n - p - 3

is a lower bound for the local dimensions of solutions of Brent equations. Moreover, the gap between the lower and upper bounds is discussed.

引用次数: 0

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