We say that two graphs G1 and G2 with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph Kn, n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there...

The $q$-Onsager algebra $O_q$ is presented by two generators $W_0$, $W_1$ and relations, called the $q$-Dolan/Grady relations. Recently Baseilhac Koizumi introduced a current $\mathcal A_q$ for $O_q$. Soon afterwards, Shigechi gave presentation of We show that these give PBW basis A_q$. Using this basis, we isomorphic to $O_q \otimes \mathbb F \lbrack z_1, z_2, \ldots \rbrack$, where $\mathbb F...

This article deenes ordered n-tuples, projections and Cartesian products for n = 5. We prove many theorems concerning the basic properties of the n-tuples and Cartesian products that may be utilized in several further, more challenging applications. A few of these theorems are a strightforward consequence of the regularity axiom. The article originated as an upgrade of the article 3]. The artic...

We study a Leech homology of a locally bounded free partially commutative monoid M(E, I). Given a contravariant natural system of abelian groups F on M(E, I) we build a precubical set T (E, I) with a homological system of abelian groups F and prove that the Leech homology groups Hn(M(E, I), F ) are isomorphic to the cubical homology groups Hn(T (E, I), F ), n > 0. As a consequence we have confi...

We define basics of (4, 4) 2D harmonic superspace with two independent sets of SU (2) harmonic variables and apply it to construct new superfield actions of (4, 4) supersymmetric two-dimensional sigma models with torsion and mutually commuting left and right complex structures, as well as of their massive deformations. We show that the generic off-shell sigma model action is the general action ...

The aim of this paper was to introduce and investigate a new seminorm operator tuples on complex Hilbert space H when an additional semi-inner product structure defined by positive (semi-definite) A is considered. We prove the equality between well-known A-joint in case A-doubly-commuting A-hyponormal operators. This study extension result [Results Math 75, 93(2020)] allows us show that followi...

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

We show that the symmetric product of a flat affine scheme over a commutative ring can be embedded into the quotient by the general linear group of the scheme of commuting matrices. We also prove that the symmetric product of the affine space is isomorphic to the above quotient when the base ring is a characteristic zero field. Over an infinite field of arbitrary characteristic the quotient of ...

Non-commutative multi-variable versions of weighted shifts arise naturally as ‘weighted’ left creation operators acting on Fock space. We investigate the unital wot-closed algebras they generate. The unweighted case yields non-commutative analytic Toeplitz algebras. The commutant can be described in terms of weighted right creation operators when the weights satisfy a condition specific to the ...