GraphTheory/IsRamanujanGraph - Maple Help
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GraphTheory

  

IsRamanujanGraph

  

test if graph is Ramanujan

 

Calling Sequence

Parameters

Options

Description

Definition

Ramanujan graphs in SpecialGraphs

Examples

Compatibility

Calling Sequence

IsRamanujanGraph(G,opts)

Parameters

G

-

graph

opts

-

(optional) equation of the form parameters=true or parameters=false

Options

• 

parameters : keyword option of the form parameters=true or parameters=false. This specifies whether the parameters [d, lambda] should be returned when the graph is Ramanujan. The default is false.

Description

• 

The IsRamanujanGraph(G) command returns true if G is a Ramanujan graph and false otherwise.

Definition

• 

An undirected graph G is Ramanujan if it is a regular graph with degree d and lambda2d1, where lambda is the maximum of the absolute values of all the eigenvalues of the adjacency matrix of G excluding the greatest eigenvalue d.

• 

These graphs are named after Srinivasa Ramanujan.

Ramanujan graphs in SpecialGraphs

• 

The following are graphs in the SpecialGraphs subpackage which are Ramanujan.

Graph

Number of Vertices

d

lambda

Petersen graph

10

3

3

Icosahedron graph

12

5

sqrt(5)

Clebsch graph

16

5

3

Complete graph

d

d-1

1

Paley graph

p

(q-1)/2

(sqrt(q)+1)/2

Examples

withGraphTheory:

withSpecialGraphs:

GGraph1,2,2,3,3,1,3,4

GGraph 1: an undirected graph with 4 vertices and 4 edge(s)

(1)

IsRamanujanGraphG

false

(2)

PPetersenGraph

PGraph 2: an undirected graph with 10 vertices and 15 edge(s)

(3)

IsRamanujanGraphP,'parameters'

true,3,2.000000000

(4)

PPaleyGraph29

PGraph 3: an undirected graph with 29 vertices and 203 edge(s)

(5)

IsRamanujanGraphP,'parameters'

true,14,3.192582400

(6)

CClebschGraph

CGraph 4: an undirected graph with 16 vertices and 40 edge(s)

(7)

IsRamanujanGraphC,'parameters'

true,5,3.000000000

(8)

Compatibility

• 

The GraphTheory[IsRamanujanGraph] command was introduced in Maple 2023.

• 

For more information on Maple 2023 changes, see Updates in Maple 2023.

See Also

Degree

DegreeSequence

IsRegular