Graph and Network Theory

  title={Graph and Network Theory},
  author={Ernesto Estrada},
This Chapter introduces the basic concepts of graphs and networks and their applications in physics. It explains the connections of graph theory with condensed matter and statistical physics by studying the tight-binding, Hubbard and Potts models. The use of graphs for solving problems in quantum field theory is illustrated by studying the Feynman graphs. Other physical scenarios in which graph theory is presented here are the study of electrical networks and vibrational systems. The Chapter… 

Communication Melting in Graphs and Complex Networks

Communicability, a topological descriptor that reveals the efficiency of the network functionality in terms of these diffusive paths, could be used to reveal the transitions mentioned, and it is shown that the communicability function plays the role of the thermal Green's function of a network of harmonic oscillators.

Using graph theory for automated electric circuit solving

A student project is described where a computational approach to electric circuit solving which is based on graph theoretic concepts is developed to reach the ambitious goal of implementing automated circuit solving.

Spectral statistics of random geometric graphs

The spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdős-Rényi, Barabási-Albert and Watts-Strogatz random graphs and finds a parameter-dependent transition between Poisson and Gaussian orthogonal ensemble statistics.

Matching number, Hamiltonian graphs and discrete magnetic Laplacians.

In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the

Deformation and Failure Onset of Random Elastic Beam Networks Generated From the Same Type of Random Graph

Deformation and failure onset of random elastic beam networks is investigated numerically. Different types of planar beam networks are generated from a random graph for which geometrical and

From Infection Clusters to Metal Clusters: Significance of the Lowest Occupied Molecular Orbital (LOMO)

The LOMO coefficient can be regarded as a manifestation of the centrality of atoms in an atomic assembly, indicating which atom plays the most important role in the assembly or which one has the greatest influence on the network of these atoms.

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

A spanning tree generating function T(z) for infinite periodic vertex-transitive (vt) lattices L vt has been proposed by Guttmann and Rogers (2012 J. Phys. A: Math. Theor. 45 494001). Their spanning



The Structure of Complex Networks: Theory and Applications

The second part of this book is devoted to the analysis of genetic, protein residue, protein-protein interaction, intercellular, ecological and socio-economic networks, including important breakthroughs as well as examples of the misuse of structural concepts.

Chemical Graph Theory

INTRODUCTION. ELEMENTS OF GRAPH THEORY. The Definition of a Graph. Isomorphic Graphs and Graph Automorphism. Walks, Trails, Paths, Distances and Valencies in Graphs. Subgraphs. Regular Graphs. Trees.

Graph theory and statistical physics

A little statistical mechanics for the graph theorist

Handbook of Graphs and Networks: From the Genome to the Internet

This book defines the field of complex interacting networks in its infancy and presents the dynamics of networks and their structure as a key concept across disciplines and offers concepts to model network structures and dynamics, focussed on approaches applicable across disciplines.

The Physics of Communicability in Complex Networks

Random graphs with arbitrary degree distributions and their applications.

It is demonstrated that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Introduction to Quantum Graphs

A "quantum graph" is a graph considered as a one-dimensional complex and equipped with a differential operator ("Hamiltonian"). Quantum graphs arise naturally as simplified models in mathematics,

Community detection in graphs