There is a large, popular, and growing literature on "scale-free" networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scale-free graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scale-free. This structural view can be related to previously studied graph properties such as the various notions of self-similarity, likelihood, betweenness and assortativity. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the current literature, and offers a rigorous and quantitative alternative, while suggesting the potential for a rich and interesting theory. This paper is aimed at readers familiar with the basics of Internet technology and comfortable with a theorem-proof style of exposition, but who may be unfamiliar with the existing literature on scale-free networks.

Single-layer single-crystalline SnSe nanosheet with four-atomic thickness of ~1.0 nm and lateral size of ~300 nm is presented here by using a one-pot synthetic method. It is found that 1,10-phenanthroline plays an important role in determining the morphology of the SnSe product as three-dimensional SnSe nanoflowers are obtained in the absence of 1,10-phenanthroline while keeping other reaction parameters the same. The evolution process study discloses that single-crystalline nanosheets are obtained from the coalescence of the SnSe nucleus in an orientated attachment mechanism. Band gap determination and optoelectronic test based on hybrid films of SnSe and poly(3-hexylthiophene) indicate the great potential of the ultrathin SnSe nanosheets in photodector and photovoltaic, and so forth.

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