A Study on Edge-Set Graphs of Certain Graphs

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th $s$-element subset of $E(G)$ and any two vertices $v_{s,i}, v_{k,m}$ of $\mathcal G_G$ are adjacent if and only if there is at least one edge in the edge-subset corresponding to $v_{s,i}$ which is adjacent to at least one edge in the edge-subset corresponding to $v_{k,m}$ where $s,k$ are positive integers. It can be noted that the edge-set graph $\mathcal G_G$ of a graph $G$ id dependent on both the structure of $G$ as well as the number of edges $\epsilon.$ We also discuss the characteristics and properties of the edge-set graphs corresponding to certain standard graphs.Comment: 10 pages, 2 figure

Governance, scale and the environment: the importance of recognizing knowledge claims in transdisciplinary arenas

Any present day approach of the world’s most pressing environmental problems involves both scale and governance issues. After all, current local events might have long-term global consequences (the scale issue) and solving complex environmental problems requires policy makers to think and govern beyond generally used time-space scales (the governance issue). To an increasing extent, the various scientists in these fields have used concepts like social-ecological systems, hierarchies, scales and levels to understand and explain the “complex cross-scale dynamics” of issues like climate change. A large part of this work manifests a realist paradigm: the scales and levels, either in ecological processes or in governance systems, are considered as “real”. However, various scholars question this position and claim that scales and levels are continuously (re)constructed in the interfaces of science, society, politics and nature. Some of these critics even prefer to adopt a non-scalar approach, doing away with notions such as hierarchy, scale and level. Here we take another route, however. We try to overcome the realist-constructionist dualism by advocating a dialogue between them on the basis of exchanging and reflecting on different knowledge claims in transdisciplinary arenas. We describe two important developments, one in the ecological scaling literature and the other in the governance literature, which we consider to provide a basis for such a dialogue. We will argue that scale issues, governance practices as well as their mutual interdependencies should be considered as human constructs, although dialectically related to nature’s materiality, and therefore as contested processes, requiring intensive and continuous dialogue and cooperation among natural scientists, social scientists, policy makers and citizens alike. They also require critical reflection on scientists’ roles and on academic practices in general. Acknowledging knowledge claims provides a common ground and point of departure for such cooperation, something we think is not yet sufficiently happening, but which is essential in addressing today’s environmental problems

Valuing information from mesoscale forecasts

The development of meso-gamma scale numerical weather prediction (NWP) models requires a substantial investment in research, development and computational resources. Traditional objective verification of deterministic model output fails to demonstrate the added value of high-resolution forecasts made by such models. It is generally accepted from subjective verification that these models nevertheless have a predictive potential for small-scale weather phenomena and extreme weather events. This has prompted an extensive body of research into new verification techniques and scores aimed at developing mesoscale performance measures that objectively demonstrate the return on investment in meso-gamma NWP. In this article it is argued that the evaluation of the information in mesoscale forecasts should be essentially connected to the method that is used to extract this information from the direct model output (DMO). This could be an evaluation by a forecaster, but, given the probabilistic nature of small-scale weather, is more likely a form of statistical post-processing. Using model output statistics (MOS) and traditional verification scores, the potential of this approach is demonstrated both on an educational abstraction and a real world example. The MOS approach for this article incorporates concepts from fuzzy verification. This MOS approach objectively weighs different forecast quality measures and as such it is an essential extension of fuzzy methods