Strongly regular graph: Wikis

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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.

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).

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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