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# Note for Graph and Network Theory - GNT by Abhishek Apoorv

• Graph and Network Theory - GNT
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Abhishek Apoorv
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Graph and Network Theory Ernesto Estrada Department of Mathematics and Statistics University of Strathclyde, Glasgow Introduction ................................................................................................................................ 2 1 The language of graphs and networks .................................................................................... 3 1.1 Graph operators ................................................................................................................ 3 1.2 General graph concepts .................................................................................................... 5 1.3 Types of graphs ................................................................................................................ 6 2 Graphs in condensed matter physics ....................................................................................... 7 2.1 Tight-binding models ....................................................................................................... 7 2.1.1 Nullity and zero-energy states ................................................................................... 9 2.2 Hubbard model ............................................................................................................... 10 3 Graphs in statistical physics .................................................................................................. 12 4 Feynman graphs .................................................................................................................... 16 4.1 Symanzik polynomials and spanning trees .................................................................... 17 4.2 Symanzik polynomials and the Laplacian matrix .......................................................... 20 4.3 Symanzik polynomials and edge deletion/contraction ................................................... 21 5 Graphs and electrical networks ............................................................................................. 21 6 Graphs and vibrations ........................................................................................................... 23 6.1 Graph vibrational Hamiltonians ..................................................................................... 24 6.2 Network of Classical Oscillators .................................................................................... 24 6.3 Network of Quantum Oscillators ................................................................................... 26 7 Random graphs ..................................................................................................................... 28 8 Introducing complex networks ............................................................................................. 30 9 Small-World networks .......................................................................................................... 32 10 Degree distributions ............................................................................................................ 34 10.1 ‘Scale-free’ networks................................................................................................... 36 11 Network motifs ................................................................................................................... 37 12 Centrality measures ............................................................................................................. 38 1

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13 Statistical mechanics of networks ....................................................................................... 41 13.1 Communicability in networks ...................................................................................... 42 14 Communities in networks ................................................................................................... 43 15 Dynamical processes on networks ...................................................................................... 45 15.1 Consensus ..................................................................................................................... 45 15.2 Synchronization in networks ........................................................................................ 47 15.3 Epidemics on networks ................................................................................................ 48 Glossary ................................................................................................................................... 50 List of works cited ................................................................................................................... 51 Further reading ......................................................................................................................... 53 Introduction Graph Theory was born in 1736 when Leonhard Euler published “Solutio problematic as geometriam situs pertinentis” (The solution of a problem relating to the theory of position) (Euler, 1736). This history is well documented (Biggs et al., 1976) and widely available in any textbook of graph or network theory. However, the word graph appeared for the first time in the context of natural sciences in 1878, when the English mathematician James J. Sylvester wrote a paper entitled “Chemistry and Algebra” which was published in Nature (Sylvester, 1877-78), where he wrote that “Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekulean diagram or chemicograph”. The use of graph theory in condensed matter physics, pioneered by many chemical and physical graph theorists (Harary, 1968; Trinajstić, 1992), is today well established; it has become even more popular after the recent discovery of graphene. There are few, if any, areas of physics in the XXIst century in which graphs and network are not involved directly or indirectly. Hence it is impossible to cover all of them in this Chapter. Thus I owe the reader an apology for the incompleteness of this Chapter and a promise to write a more complete treatise. For instance, quantum graphs are not considered in this Chapter and the reader is referred to a recent introductory monograph on this topic for details (Berkolaiko, Kuchment, 2013). In this chapter we will cover some of the most important areas of applications of graph theory in physics. These include condensed matter physics, statistical physics, quantum electrodynamics, electrical networks and vibrational problems. In the second part we summarise some of the most important aspects of the study of complex networks. This is an interdisciplinary area which has emerged with tremendous impetus in the XXIst century which studies networks appearing in complex systems. These systems range from molecular and biological to ecological, social and technological systems. Thus graph theory and network theory have helped to broaden the horizons of physics to embrace the study of new complex systems. We hope this chapter motivates the reader to find more about the connections between graph/network theory and physics, consolidating this discipline as an important part of the curriculum for the physicists of the XXIst century. 2