This paper is devoted to new algebraic structures, called qualgebras and squandles. Topologically, they emerge as an algebraic counterpart of knotted 3-valent graphs, just like quandles can be seen as an "algebraization" of knots. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. We discuss basic properties of these structures, and introduce and study the notions of qualgebra/squandle 2-cocycles and 2-coboundaries. Knotted 3-valent graph invariants...

We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_N$-spaces needed to represent (up to a constant $C>1$) the $M_N$-version of the $n$-dimensional operator Hilbert space $O{H}_n$ as a direct sum of copies of $M_N$. We show that, when $C$ is close to 1, this multiplicity grows as $exp\beta n{N}^2$ for...

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements about matrix models.

The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a $\lambda$-ideal of the special $\lambda$-ring of symmetric group class functions.

The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q/G$ if and only if there exists a normalized left transversal $\Sigma \subset Q$ to $G$ in $Q$ such that the right translations by elements of $\Sigma$ commute with all right translations by elements of the subgroup $G$. Moreover, a loop $Q$ is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and there exists...

We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.

A vertex subset S of a graph G is a perfect (resp. quasiperfect) dominating set in G if each vertex v of G∖S is adjacent to only one vertex ($d_v$ ∈ 1,2 vertices) of S. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol 3,6 and in its toroidal quotients are investigated, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets S with induced components of the form $K_{\nu }$, where ν ∈ 1,2,3 depends only on S.

The Sombor index $SO\left(G\right)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$ of all edges $uv$ of $G$, where $d_G\left(u\right)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G=(V,E)$ is called a quasi-tree if there exists $u\in V\left(G\right)$ such that $G-u$ is a tree. Denote $𝒬(n,k)=\{G:G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G\left(u\right)=k\}$. We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬(n,k)$. Furthermore, we characterized the corresponding extremal graphs, respectively.

La récente découverte des “quasicristaux” et leurs liens avec les pavages de Penrose ont entraîné un regain d'intérêt pour les pavages apériodiques du plan. Nous montrons ici que le pavage régulier de Robinson est engendré par un automate fini bidimensionnel, et qu'il donne une généralisation à deux dimensions du pliage de papier.