We derive a set of algebraic equations, the so-called multipartite separability eigenvalue equations. Based on their solutions, we introduce a universal method for the construction of multipartite entanglement witnesses. We witness multipartite entanglement of 10(3) coupled quantum oscillators, by solving our basic equations analytically. This clearly demonstrates the feasibility of our method ...

Let \((X, d)\) be a semimetric space and let \(G\) graph. We say that is the diametrical graph of if \(X\) vertex set adjacency vertices \(x\) \(y\) equivalent to equality \(\diam X = d(x, y)\). It shown with diameter \(d^*\) ultrametric iff d_{\varepsilon})\) \(d_{\varepsilon}(x, y) \min\{d(x, y), \varepsilon\}\) complete multipartite for every \(\varepsilon \in (0, d^*)\). A refinement last r...

Abstract PQ-type adjacency polytopes $$\nabla ^PQ _G$$ ∇ P Q G are lattice arising from finite graphs G . There is a connection between and the engineering problem known as power-flow study,...

We study the relation between distillability of multipartite states and violation of Bell's inequality. We prove that there exist multipartite bound entangled states (i.e., nonseparable, nondistillable states) that violate a multipartite Bell inequality. This implies that (i) violation of Bell's inequality is not a sufficient condition for distillability and (ii) some bound entangled states can...

A total coloring of a graph G is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. The concept of total coloring was introduced by V. Vizing [15] and independently by M. Behzad [3]. The total chromatic number χ (G) is the smallest number of colors needed for total coloring of G. In 1965 V. Vizing and M. Behzad c...

The anti-adjacency matrix of a graph is constructed from the distance by keeping each row and column only largest distances. This can be interpreted as opposite adjacency matrix, which instead in distances equal to 1. (anti-)adjacency eigenvalues are those its matrix. Employing novel technique introduced Haemers (2019) [9], we characterize all connected graphs with exactly one positive eigenval...

Large graph-based datasets are common to many applications because of the additional structure provided to data by graphs. Patterns extracted from graphs must adhere to these structural properties, making them a more complex class of patterns to identify. The role of graph mining is to efficiently extract these patterns and quantify their significance. In this thesis, we focus on two applicatio...

We construct a class of multipartite states possessing rotational SO(3) symmetry – these are states of K spin-jA particles and K spin-jB particles. The construction of symmetric states follows our two recent papers devoted to unitary and orthogonal multipartite symmetry. We study basic properties of multipartite SO(3) symmetric states: separability criteria and multi-PPT conditions.

The competition graph of a digraphD is a graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x inD such that (u, x) and (v, x) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number ...

For a connected graph G and point v of G, let G, be the subgraph induced by the points adjacent to v . This G is called locally Go if G, = Go for all points v of G. A graph is called locally homogeneous if it is locally Go for some Go. Recent work concerning local homogeneity has focused on two broad questions. First, for which Go does there exist a G that is locally Go? This has been settled f...