# 6. The Linear Complexity of a Graph

- Matthias Dehmer
^{3}, - Abbe Mowshowitz
^{4}and - Frank Emmert-Streib
^{4}

Published Online: 12 JUL 2013

DOI: 10.1002/9783527670468.ch06

© 2013 Wiley-VCH Verlag GmbH & Co. KGaA

Book Title

## Advances in Network Complexity

Additional Information

#### How to Cite

Neel, D. L. and Orrison, M. E. (2013) The Linear Complexity of a Graph, in Advances in Network Complexity (eds M. Dehmer, A. Mowshowitz and F. Emmert-Streib), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. doi: 10.1002/9783527670468.ch06

#### Editor Information

- 3
UMIT, Institut für Bioinformatik und, Translationale Forschung, Eduard-Wallnöfer-Zentrum 1, 6060 Hall in Tyrol, Austria

- 4
The City College of New York, Department of Computer Science, 138th Street at Convent Avenue, New York, NY 10031, USA

#### Publication History

- Published Online: 12 JUL 2013
- Published Print: 10 JUL 2013

#### Book Series:

#### ISBN Information

Print ISBN: 9783527332915

Online ISBN: 9783527670468

- Summary
- Chapter
- References

### Keywords:

- adjacency matrix;
- graph;
- linear complexity;
- lower bound;
- upper bound

### Summary

This chapter deals with the measurement of graph complexity by focusing on the adjacency matrix of the graph, and the linear complexity of the matrix. The first section of the chapter describes the linear complexity of a matrix, and introduces the notion of the linear complexity of a graph. The second section explores some properties of the irreducible subgraph. The third section presents several upper and lower bounds on the linear complexity of a graph. In the fourth section, the author considers the linear complexity of several well-known classes of graphs. The final section offers an upper bound for the linear complexity of a graph that is based on the use of clique partitions. All graphs considered in the chapter are finite, simple, and undirected. Comparison between linear complexity and other common methods of measuring graph complexity helps to determine which aspects of complexity are being well captured.