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From Wikipedia, the free encyclopedia

The Paley graph of order 13, a strongly regular graph with parameters srg(13,6,2,3).
Graph families defined by their automorphisms
distance-transitive \rightarrow distance-regular \leftarrow strongly regular
\downarrow
symmetric (arc-transitive) \leftarrow t-transitive, t ≥ 2
\downarrow(if connected)
vertex- and edge-transitive \rightarrow edge-transitive and regular \rightarrow edge-transitive
\downarrow \downarrow
vertex-transitive \rightarrow regular
\uparrow
Cayley graph skew-symmetric

Let G = (V,E) be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that:

  • Every two non-adjacent vertices have μ common neighbours.

A graph of this kind is sometimes said to be an srg(v,k,λ,μ).

Some authors exclude[citation needed] graphs which satisfy the definition trivially, namely those graphs which are the disjoint union of one or more equal-sized complete graphs, and their complements, the Turán graphs.

A strongly regular graph is a distance-regular graph with diameter 2, but only if μ is non-zero.

Contents

Properties

  • The four parameters in an srg(v,k,λ,μ) are not independent, as it is easy to show that:
(vk − 1)μ = k(k − λ − 1)
  • Let I denote the identity matrix (of order v) and let J denote the matrix whose entries all equal 1. The adjacency matrix A of a strongly regular graph satisfies these properties :
    • A\times J = kJ
      (This is a trivial restatement of the vertex degree requirement).
    • A2 + (μ − λ)A + (μ − k)I = μJ
      (The first term gives the number of 2-step paths from each vertex to all vertices. For the vertex pairs directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to λ. For the vertex pairs not directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to μ. For the trivial self-pairs, the equation reduces to the degree being equal to k).
  • The graph has exactly three eigenvalues:
    • k whose multiplicity is 1
    • \frac{1}{2}\left[(\lambda-\mu)+\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}\right] whose multiplicity is \frac{1}{2} \left[(v-1)-\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}}\right]
    • \frac{1}{2}\left[(\lambda-\mu)-\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}\right] whose multiplicity is \frac{1}{2} \left[(v-1)+\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}}\right]
  • Strongly regular graphs for which 2k + (v − 1)(λ − μ) = 0 are called conference graphs because of their connection with symmetric conference matrices. Their parameters reduce to srg\left(v, \frac{v-1}{2}, \frac{v-5}{4}, \frac{v-1}{4}\right).
  • Strongly regular graphs for which 2k+(v-1)(\lambda-\mu) \ne 0 have integer eigenvalues with unequal multiplicities.
  • The complement of an srg(v,k,λ,μ) is also strongly regular. It is an srg(v, v−k−1, v−2−2k+μ, v−2k+λ).

Examples

See also

Notes

References

  • A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5
  • Chris Godsil and Gordon Royle (2004), Algebraic Graph Theory. New York: Springer-Verlag. ISBN 0-387-95241-1

External links


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