Linear Algebra and its Applications最新文献

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Hoffman colorings of graphs

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.036

Aida Abiad , Wieb Bosma , Thijs van Veluw

Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for general graphs not much is known. We investigate tightness of the Hoffman bound, with a particular focus on irregular graphs, obtaining several results on the graph structure of Hoffman colorings. In particular, we prove a Decomposition Theorem, which characterizes the structure of Hoffman colorings, and we use it to completely classify Hoffman colorability of cone graphs and line graphs. We also prove a partial converse, the Composition Theorem, leading to an algorithm for computing all connected Hoffman colorable graphs for some given number of vertices and colors. Since several graph coloring parameters are known to be sandwiched between the Hoffman bound and the chromatic number, as a byproduct of our results, we obtain the values of these chromatic parameters.

引用次数: 0

Numerical range of real-valued linear mapping on the complex Stiefel manifold: Convexity and application

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.031

Hanzhi Chen, Zhenhong Huang, Mengmeng Song, Yong Xia

The study confirms the convexity of the joint numerical range of any k real-valued linear functions on the

n \times p

complex Stiefel manifold under the condition

k \leq 2 n - 2 p + 1

. Revealing the hidden convexity of fractional linear programming on the complex Stiefel manifold, a first-time study, serves as an impactful application.

引用次数: 0

Spectral extremal graphs for disjoint odd wheels

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.034

Yu Luo , Zhenyu Ni , Yanxia Dong

For a given graph F, let

ex (n, F)

and

spex (n, F)

be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all F-free graphs of order n, respectively.

EX (n, F)

and

SPEX (n, F)

consist of the extremal graphs associated with

ex (n, F)

and

spex (n, F)

, respectively. The odd wheel

W_{2 k + 1}

is constructed by joining a vertex to a cycle

C_{2 k}

. Cioabă, Desai and Tait determined the spectral extremal graphs of

W_{2 k + 1}

for

k \geq 2, k \notin {4, 5}

. Xiao and Zamora determined the Turán number and all extremal graphs for

t W_{2 k + 1}

, which is the union of t vertex-disjoint copies of

W_{2 k + 1}

for

k \geq 3

. In this paper, we focus on the graph with maximum spectral radius among those that exclude any subgraph isomorphic to

t W_{2 k + 1}

. We present structural characteristics of these spectral extremal graphs for

k \geq 3, k \notin {4, 5}

. Furthermore, we demonstrate that

SPEX (n, t W_{2 k + 1}) \cap EX (n, t W_{2 k + 1}) = \emptyset

for

k \geq 10

and n large enough.

引用次数: 0

Constructing adjacency and distance cospectral graphs via regular rational orthogonal matrix

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.033

Lihuan Mao , Yuanhang Xu , Fenjin Liu , Bei Liu

Two graphs G and H are cospectral if they share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature. In this paper, we construct infinite families of adjacency cospectral graphs through the GM-switching method based on generalized Johnson graphs. We give some graph operations (e.g. rooted-product, corona, cartesian product, and coalescence) to construct distance cospectral graphs with different edges via a regular rational orthogonal matrix.

引用次数: 0

Polynomial codimension growth of superalgebras with superautomorphism

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.035

Sara Accomando

In this paper we present some results concerning associative superalgebras endowed with a superautomorphism of order ≤2. We characterize the superalgebras with superautomorphism with multiplicities of the cocharacter bounded by a constant. Moreover, we determine the characterization of the superalgebras with superautomorphism with polynomial growth of the codimensions and we give a classification of the subvarieties of the varieties of almost polynomial growth. Finally, we characterize the superalgebras with superautomorphism with linear growth of the codimensions.

引用次数: 0

Bipartite q-Kneser graphs and two-generated irreducible linear groups

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.032

S.P. Glasby , Alice C. Niemeyer , Cheryl E. Praeger

<div><div>Let <math><mi>V</mi><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup></math> be a d-dimensional vector space over the field <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> of order q. Fix positive integers <math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math> satisfying <math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mi>d</mi></math>. Motivated by analysing a fundamental algorithm in computational group theory for recognising classical groups, we consider a certain quantity <math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> which arises in both graph theory and group representation theory: <math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> is the proportion of 3-walks in the ‘bipartite q-Kneser graph’ <math><msub><mrow><mi>Γ</mi></mrow><mrow><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> that are closed 3-arcs. We prove that, for a group G satisfying <math><msub><mrow><mi>SL</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>⊴</mo><mi>G</mi><mo>⩽</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>, the proportion of certain element-pairs in G called ‘<math><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>-stingray duos’ which generate an irreducible subgroup is also equal to <math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>. We give an exact formula for <math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>, and prove that<math><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo><</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo><</mo><mn>1</mn

引用次数: 0

The Rao-Mitra-Bhimasankaram relation is strongly antisymmetric

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-27 DOI: 10.1016/j.laa.2025.01.029

Oskar Maria Baksalary , Dennis Bernstein

For

n \times m

complex matrices A and B, the Rao-Mitra-Bhimasankaram (RMB) relation

\overset{RMB}{\leq}

, defined by

A \overset{RMB}{\leq} B

A A^{⁎} A = A B^{⁎} A

, is reflexive and antisymmetric, but not transitive. This paper shows that, despite the lack of transitivity,

\overset{RMB}{\leq}

is strongly antisymmetric in the sense that, for all integers

n \geq 2

A_{1} \overset{RMB}{\leq} \dots \overset{RMB}{\leq} A_{n} \overset{RMB}{\leq} A_{1}

implies

A_{1} = \dots = A_{n}

. The proof of this result is based on a novel proof that

\overset{RMB}{\leq}

is antisymmetric.

引用次数: 0

Rosenbrock's theorem on system matrices over elementary divisor domains

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.028

Froilán M. Dopico , Vanni Noferini , Ion Zaballa

Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix G with the Smith form of an irreducible polynomial system matrix P giving rise to G and the Smith form of a submatrix of P. This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock's theorem to system matrices P with entries in an arbitrary elementary divisor domain

R

and matrices G with entries in the field of fractions of

R

. These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock's theorem can be extended when the system matrix P itself has entries in the field of fractions of the elementary divisor domain.

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引用次数: 0

When does subtracting a rank-one approximation decrease tensor rank?

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.025

Emil Horobeţ , Ettore Teixeira Turatti

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.

引用次数: 0

On the nullity of middle graphs

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.030

Xinmei Yuan , Danyi Li , Weigen Yan

Let G be a connected graph, and let

L (G)

and

M (G)

be the line graph and middle graph of G. Gutman and Sciriha (On the nullity of line graphs of trees, Discrete Mathematics, 232 (2001), 35-45) proved that the nullity

η (L (T))

L (T)

of a tree T satisfies

η (L (T)) = 0

η (L (T)) = 1

. But the problem to determine which trees T satisfy

η (L (T)) = 0

η (L (T)) = 1

is still open. In this paper, we prove that

η (M (G)) = 1

if G is a bipartite graph, and

η (M (G)) = 0

otherwise. As an application, we show that

η (G (n, m)) = 1

for the so-called silicate network

G (n, m)

obtained from the hexagonal lattice in the context of statistical physics.

引用次数: 0

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