Due to its numerous important applications[1–5] in the area of quantum information and computation, entanglement has been extensively studied in the past decade. The concept of considering entanglement as a new type of useful resource has been widely accepted. Efforts have been made to quantitatively and qualitatively measure this kind of resource in both bipartite and multipartite cases[6–8]. ...

Let k be an integer. Two vertex k-colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k-mixing if any proper k-coloring can be transformed into any other through a sequence of adjacent proper k-colorings. Jerrum proved that any graph is k-mixing if k is at least the maximum degree plus two. We first improve Jerrum’s bound using the grundy number, which is the wors...

Abstract The notion of a word-representable graph has been studied in a series of papers in the literature. A graph G = (V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E. If V = {1, . . . , n}, this is equivalent to saying that G is wordrepresentable if for all x, y ∈ {1, . . . , n}, xy ∈ E if and ...

Comparability between different public examinations in the same and also dzfferent subjects, has been a continuing requirement in the UK. There is a current renewed interest in between-subject comparability, especially at A-level. The present paper examines the assumptions behind attempts to achieve comparability and explores the educational implications of some of the statistical procedures wh...

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

‎In this paper we prove that a simplicial complex is determined‎ ‎uniquely up to isomorphism by its barycentric subdivision as well as‎ ‎its comparability graph‎. ‎We also put together several algebraic‎, ‎combinatorial and topological invariants of simplicial complexes‎.