We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabil...

In this paper, we define the non-centralizer graph associated to a finite group G, as the graph whose vertices are the elements of G, and whose edges are obtained by joining two distinct vertices if their centralizers are not equal. We denote this graph by ΥG. The non-centralizer graph is used to study the properties of the non-commuting graph of an AC-group. We prove that the non-centralizer g...

‎in a recent paper c‎. ‎miguel proved that the diameter of the commuting graph of the matrix ring $mathrm{m}_n(mathbb{r})$ is equal to $4$ if either $n=3$ or $ngeq5$‎. ‎but the case $n=4$ remained open‎, ‎since the diameter could be $4$ or $5$‎. ‎in this work we close the problem showing that also in this case the diameter is $4$.

The commuting graph of a group G, denoted by Γ(G), is a simple graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. In [1] it is conjectured that if M is a simple group and G is a group satisfying Γ(G) ∼= Γ(M), then G ∼= M . In this paper we prove this conjecture for many simple groups.

The commuting graph of a group G, denoted by Γ(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Zm be the commutative ring of equivalence classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Zm) to contribute to the conje...

let $g$ be a non-abelian group of order $p^n$‎, ‎where $nleq 5$ in which $g$ is not extra special of order $p^5$‎. ‎in this paper we determine the maximal size of subsets $x$ of $g$‎ ‎with the property that $xyneq yx$ for any $x,y$ in $x$ with‎ ‎$xneq y$‎.

The commuting graph $$\varDelta (G)$$ of a finite non-abelian group G is simple with vertex set G, and two distinct vertices x, y are adjacent if $$xy = yx$$ . In this paper, first we discuss some properties We determine the edge connectivity minimum degree prove that both equal. Then, other invariants, namely: matching number, clique boundary vertex, studied. Also, give necessary sufficient ...

Let G be a group and X a subset of G. The commuting graph on X, denoted C(G,X), has vertex set X and an edge joining x, y ∈ X whenever xy = yx. If in addition X is a set of involutions, then C(G,X) is called a commuting involution graph. Commuting graphs have been investigated by many authors. Sometimes they are tools used in the proof of a theorem, or they may be studied as a way of shedding l...