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6950 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 47 of 348

  • 7 - Transforming Graphs

  • from Part II - Applications
  • Michael Rudolph
  • Book:
    The Mathematics of Finite Networks
    Published online:
    30 April 2022
    Print publication:
    12 May 2022, pp 291-324

  • Summary

    Navigating across sometimes treacherous waters, we demonstrated in the previous chapter the construction and utilisation of a small selection of graph observables for measuring various properties of finite random graphs. Together with the generation of operator representations for arbitrary graph models, we should now be in possession of a sufficiently equipped toolset with which to further explore and characterise on rigorous algebraical grounds the plethora of graph models in the applied graph-theoretical literature. However, our adventurous journey would not be complete without touching upon another crucial aspect exhibited by many real-world networks - their dynamic nature. In this final chapter, we will explore with one hopefully light-hearted, playful example - the game of chess - how to formulate such dynamical aspects in our operator graph-theoretical language. As we will witness here, the construction of graphs that describe possible moves of chess pieces at any position during a game, and the transformations that lead to changes of such positional chess graphs, pose a formidable challenge for not only computational algorithms.

  • 5 - Generating Graphs

  • from Part II - Applications
  • Michael Rudolph
  • Book:
    The Mathematics of Finite Networks
    Published online:
    30 April 2022
    Print publication:
    12 May 2022, pp 181-237

  • Summary

    The first part of this book led us on a journey from one of the undoubtedly most cherished fields of applied mathematics, classical graph theory, across the ghastly depths of an inherently dynamic formalisation of physical reality in terms of mappings and operators, to an inspired attempt at a fusion of both of these perspectives. With the backing of a conceptional and notational framework at hand, it is now time to put this attempt to a test, as we continue our adventurous journey with an excursion into the endless realm of applications. This second part of our journey will start in this chapter with an exploration of graph generators and their operator graph-theoretical formulation. In this undertaking, we will focus primarily on the generation of random graphs as such models enjoy, in one way or another, widespread and prominent employment throughout almost all fields of science and technology. Only the last section will see the exemplary generation of an exact graph model, the finite square grid graph, as preparation for a closer inspection of an intriguing yet unsolved problem at the very heart of condensed matter physics in the next chapter.

Page 47 of 348