A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms can preserve it. The number $G$, denoted by $D(G)$, the minimum colors required for such coloring, and threshold $\theta(G)$, $k$ that every $k$-coloring distinguishing. As an alternative definition, $\theta(G)$ one more than maximum cycles in cycle decomposition $G$. In this paper, we characterize $\thet...

Abstract Many innovations that are implemented in schools initially successful, but fail to become part of the schools’ habits and routines. Relatively little research has followed for a long(er) time. In addition, few reforms last long enough be studied longitudinally. this exploratory study, authors aim find way quantitatively measure degree sustainability data use intervention three eight ye...

Owing to nonlinear viscoelasticity, materials often show characteristic features that resemble those of thixotropy. This issue has been debated in the literature over past several decades, and experimental protocols have proposed distinguish thixotropy from viscoelasticity. In this work, we assess these by carrying out experiments using polymer solutions, thixotropic clay dispersions, modeling ...

In this paper, we study a new coloring parameter of graphs called the gap vertexdistinguishing edge coloring. It consists in an edge-coloring of a graph G which induces a vertex distinguishing labeling of G such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distingu...