# Graph and Network Theory

@inproceedings{Estrada2015GraphAN, title={Graph and Network Theory}, author={Ernesto Estrada}, year={2015} }

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…

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## 17 Citations

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## References

SHOWING 1-10 OF 70 REFERENCES

### The Structure of Complex Networks: Theory and Applications

- Art
- 2011

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

- Chemistry, Mathematics
- 1992

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.…

### Statistical-mechanical approach to subgraph centrality in complex networks

- Computer Science
- 2007

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

- Art
- 2003

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.

### Random graphs with arbitrary degree distributions and their applications.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2001

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

- Physics
- 2012

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,…