Linear Algebra and its Applications最新文献

A graph G is k-factor-critical if $G-S$ has a perfect matching for any subset S of $V\left(G\right)$ with $\left|S\right|=k$. An integer k-matching of G is a function $h:E\left(G\right)\to \{0,1,\dots ,k\}$ satisfying ${\sum }_{e\in \Gamma \left(v\right)}h\left(e\right)\le k$ for all $v\in V\left(G\right)$, where $\Gamma \left(v\right)$ is the set of edges incident with v. An integer k-matching h of G is called perfect if ${\sum }_{e\in E\left(G\right)}h\left(e\right)=k\left|V\right(G\left)\right|/2$. A graph G has the strong parity property if for every subset S of $V\left(G\right)$ with even size, G has a spanning subgraph F with minimum degree at least one such that $d_F\left(v\right)\equiv 1\left(\mathrm{mod}2\right)$ for all $v\in S$ and $d_F\left(u\right)\equiv 0\left(\mathrm{mod}2\right)$ for all $u\in V\left(G\right)﹨S$. In this paper, we provide edge number and spectral conditions for the k-factor-criticality, perfect integer k-matching and strong parity property of a graph, respectively.

The statement of the Karpelevič theorem concerning the location of the eigenvalues of stochastic matrices in the complex plane (known as the Karpelevič region) is long and complicated and his proof methods are, at best, nebulous. Fortunately, an elegant simplification of the statement was provided by Ito—in particular, Ito's theorem asserts that the boundary of the Karpelevič region consists of arcs whose points satisfy a polynomial equation that depends on the endpoints of the arc. Unfortunately, Ito did not prove his version and only showed that it is equivalent.

More recently, Johnson and Paparella showed that points satisfying Ito's equation belong to the Karpelevič region. Although not the intent of their work, this initiated the process of proving Ito's theorem and hence providing another proof of the Karpelevič theorem.

The purpose of this work is to continue this effort by showing that an arc appears in the prescribed sector. To this end, it is shown that there is a continuous function $\lambda :[0,1]⟶C$ such that $P^I\left(\lambda \right(\alpha \left)\right)=0$, $\forall \alpha \in [0,1]$, where $P^I$ is a Type I reduced Ito polynomial. It is also shown that these arcs are simple. Finally, an elementary argument is given to show that points on the boundary of the Karpelevič region are extremal whenever $n>3$.

In 1992, Godsil and Hensel published a ground-breaking study of distance-regular antipodal covers of the complete graph that, among other things, introduced an important connection with equi-isoclinic subspaces. This connection seems to have been overlooked, as many of its immediate consequences have never been detailed in the literature. To correct this situation, we first describe how Godsil and Hensel's machine uses representation theory to construct equi-isoclinic tight fusion frames. Applying this machine to Mathon's construction produces $q+1$ equi-isoclinic planes in $R^{q+1}$ for any even prime power $q>2$. Despite being an application of the 30-year-old Godsil–Hensel result, infinitely many of these parameters have never been enunciated in the literature. Following ideas from Et-Taoui, we then investigate a fruitful interplay with complex symmetric conference matrices.

We prove the inequality ${\lambda }_k\le d_k+d_{k-1}$ for all the eigenvalues ${\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_n$ of the Kirchhoff matrix K of a finite simple graph or quiver with vertex degrees $d_1\le d_2\le \cdots \le d_n$ and assuming $d_0=0$. Without multiple connections, the inequality ${\lambda }_k\ge \max (0,d_k-(n-k\left)\right)$ holds. A consequence in the finite simple graph or multi-graph case is that the pseudo determinant $\mathrm{Det}\left(K\right)$ counting the number of rooted spanning trees has an upper bound $2^n{\prod }_{k=1}^n{d}_k$ and that $\det (1+K)$ counting the number of rooted spanning forests has an upper bound ${\prod }_{k=1}^n(1+2d_k)$.

The main purpose of this paper is to study the class of Lie-admissible algebras $(A,.)$ such that its product is a biderivation of the Lie algebra $(A,[,\left]\right)$, where $[,]$ is the commutator of the algebra $(A,.)$. First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Lie algebras, symmetric Leibniz algebras, Lie-admissible left (or right) Leibniz algebras, Milnor algebras, and LR-algebras. Then, we establish results on the structure of these algebras in the case that the underlying Lie algebras are perfect (in particular, semisimple Lie algebras). In addition, we then study flexible $A_{BD}$-algebras, showing in particular that these algebras are extensions of Lie algebras in the category of flexible $A_{BD}$-algebras. Finally, we study left-symmetric $A_{BD}$-algebras, in particular we are interested in flat pseudo-Euclidean Lie algebras where the associated Levi-Civita products define $A_{BD}$-algebras on the underlying vector spaces of these Lie algebras. In addition, we obtain an inductive description of all these Lie algebras and their Levi-Civita products (in particular, for all signatures in the case of real Lie algebras).

A unified framework for the Expander Mixing Lemma for irregular graphs using adjacency eigenvalues is presented, as well as two new versions of it. While the existing Expander Mixing Lemmas for irregular graphs make use of the notion of volume (the sum of degrees within a vertex set), we instead propose to use the Perron eigenvector entries as vertex weights, which is a way to regularize the graph. This provides a new application of weight partitions of graphs. The new Expander Mixing Lemma versions are then applied to obtain several eigenvalue bounds for NP-hard parameters such as the zero forcing number, the vertex integrity and the routing number of a graph.

Huang, McKinnon, and Satriano conjectured that if $v\in R^n$ has distinct coordinates and $n\ge 3$, then a hyperplane through the origin other than ${\sum }_i{x}_i=0$ contains at most $2\lfloor n/2\rfloor (n-2)!$ of the vectors obtained by permuting the coordinates of v. We prove this conjecture.

In this work we consider generic losses of rank for complex valued matrix functions depending on two parameters. We give theoretical results that characterize parameter regions where these losses of rank occur. Our main results consist in showing how following an appropriate smooth SVD along a closed loop it is possible to monitor the Berry phases accrued by the singular vectors to decide if –inside the loop– there are parameter values where a loss of rank takes place. It will be needed to use a new construction of a smooth SVD, which we call the “joint-MVD” (minimum variation decomposition).

We establish novel two-sided bounds for the tracial seminorm of multilinear Schur multipliers that tighten previously known bounds. The result is obtained by a newly developed method based on polynomial chaoses.

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees $2n<250$, specifically, the question remains open only 7 values of n.