Book contents
- Frontmatter
- Contents
- Introduction
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Strongly regular graphs with no triangles
- 5 Polarities of designs
- 6 Extension of graphs
- 7 Codes
- 8 Cyclic codes
- 9 Threshold decoding
- 10 Reed-Muller codes
- 11 Self-orthogonal codes and designs
- 12 Quadratic residue codes
- 13 Symmetry codes over GF(3)
- 14 Nearly perfect binary codes and uniformly packed codes
- 15 Association schemes
- References
- Index
4 - Strongly regular graphs with no triangles
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Introduction
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Strongly regular graphs with no triangles
- 5 Polarities of designs
- 6 Extension of graphs
- 7 Codes
- 8 Cyclic codes
- 9 Threshold decoding
- 10 Reed-Muller codes
- 11 Self-orthogonal codes and designs
- 12 Quadratic residue codes
- 13 Symmetry codes over GF(3)
- 14 Nearly perfect binary codes and uniformly packed codes
- 15 Association schemes
- References
- Index
Summary
In a graph, a path is a sequence of vertices in which consecutive vertices are adjacent, and a circuit is a path with initial and terminal point equal. A graph is connected if any two vertices lie in a path. The function d defined by d(p, q) = length of the shortest path containing p and q, is a metric in a connected graph; the diameter of the graph is the largest value assumed by d. The girth of a graph is the length of the shortest circuit which contains no repeated edges, provided such a circuit exists.
For strongly regular graphs, the connectedness, diameter and girth are simply determined by the parameters. Γ is connected with diameter 2 if d > 0, and is disconnected if d = 0. Γ has a girth provided a > 1; the girth is 3 if c > 0, 4 if c = 0 and d > 1, and 5 if c = 0, d = 1.
It is easy to see that a graph with diameter 2 and maximal valency a has at most a2 + 1 vertices; and a graph with girth 5 and minimal valency a has at least a2 + 1 vertices. Equality holds in either case if and only if the graph is strongly regular with c = 0, d = 1. Such a graph is called a Moore graph with diameter 2.
- Type
- Chapter
- Information
- Graph Theory, Coding Theory and Block Designs , pp. 25 - 32Publisher: Cambridge University PressPrint publication year: 1975