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GraphTheory[SpecialGraphs]

  

PayleyGraph

  

construct Payley graph

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

PayleyGraph(p)

PayleyGraph(p,k)

PayleyGraph(p,k,m)

Parameters

p

-

prime integer

k

-

positive integer

m

-

irreducible univariate polynomial of degree k over GF(p)

Description

• 

If the input is PayleyGraph(p) then the output is an undirected unweighted simple graph G on p vertices labeled 0,1,...,p-1 where the edge {i,j}, with i<j, is in G iff j-i is a quadratic residue in Zp.

• 

If the input is PayleyGraph(p,k) then the output is an undirected unweighted simple graph G on q&equals;pk vertices labeled 0,1,...,q-1 where the edge {i,j}, i<j, is in G iff y-x is a square the finite field GF(q) where x is the ith element and y is the jth element in GF(q). The numbering of the elements in GF(q) is lexicographical.

• 

The vertex label for the element f(x) in Zp[x] is f(p). For example, in GF(23) the elements are ordered 0&comma;1&comma;x&comma;x&plus;1&comma;x2&comma;x2&plus;1&comma;x2&plus;x&comma;x2&plus;x&plus;1. Thus the label for element x2&plus;1 is 22+1=5.

• 

The field can be specified by specifying the extension polynomial by the user by specifying the extension polynomial for GF(q), a monic irreducible polynomial m(x) in Zp[x] of degree k.

Examples

withGraphTheory&colon;

withSpecialGraphs&colon;

PPayleyGraph5

P:=Graph 1: an undirected unweighted graph with 5 vertices and 5 edge(s)

(1)

EEdgesP

E:=0&comma;1&comma;0&comma;4&comma;1&comma;2&comma;2&comma;3&comma;3&comma;4

(2)

PPayleyGraph2&comma;2

P:=Graph 2: an undirected unweighted graph with 4 vertices and 6 edge(s)

(3)

EEdgesP

E:=0&comma;1&comma;0&comma;2&comma;0&comma;3&comma;1&comma;2&comma;1&comma;3&comma;2&comma;3

(4)

PPayleyGraph3&comma;2&comma;y2&plus;1

P:=Graph 3: an undirected unweighted graph with 9 vertices and 18 edge(s)

(5)

EEdgesP&colon;

See Also

SpecialGraphs

 


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