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Qualgebras and knotted 3-valent graphs

Victoria Lebed (2015)

Fundamenta Mathematicae

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...

Quantum expanders and geometry of operator spaces

Gilles Pisier (2014)

Journal of the European Mathematical Society

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 β n N 2 for...

Quasiperfect domination in triangular lattices

Italo J. Dejter (2009)

Discussiones Mathematicae Graph Theory

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 ν , where ν ∈ 1,2,3 depends only on S.

Quasi-tree graphs with the minimal Sombor indices

Yibo Li, Huiqing Liu, Ruiting Zhang (2022)

Czechoslovak Mathematical Journal

The Sombor index S O ( G ) of a graph G is the sum of the edge weights d G 2 ( u ) + d G 2 ( v ) of all edges u v of G , where d G ( u ) denotes the degree of the vertex u in G . A connected graph G = ( V , E ) is called a quasi-tree if there exists u ∈ V ( G ) 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 ( u ) = 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.

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