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networks

  

rankpoly

  

(Whitney) rank polynomial of an undirected graph

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

rankpoly(G, x, y)

Parameters

G

-

undirected graph or network

x

-

rank variable in rank poly

y

-

corank variable in rank poly

Description

• 

Important: The networks package has been deprecated.Use the superseding command GraphTheory[RankPolynomial] instead.

• 

When nG = number of vertices, mG = number of edges, and cG = number of components, one defines rank(G) = n(G) - c(G) and corank(G) = m(G) - rank(G).

• 

The rank polynomial is a sum over all subgraphs H of G of x^(rank(G) - rank(H)) y^corank(H).

• 

The coefficient of xiyj in the rank polynomial is thus the number of spanning subgraphs of G having i more components than G and having a cycle space of dimension j.

• 

This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[rankpoly](...).

Examples

Important: The networks package has been deprecated.Use the superseding command GraphTheory[RankPolynomial] instead.

withnetworks:

Gcomplete4:

rankpolyG,x,y

x3+y3+6x2+4xy+6y2+15x+15y+16

(1)

rankpolyG,1,1

64

(2)

rankpolyG,1,0

38

(3)

rankpolyG,0,1

38

(4)

rankpolyG,0,0

16

(5)

See Also

GraphTheory

GraphTheory[RankPolynomial]

networks(deprecated)[acycpoly]

networks(deprecated)[flowpoly]

networks(deprecated)[tuttepoly]

with