The Cycle Space Over A Field or Commutative Ring
The construction of the integral cycle space can be carried out for any field, abelian group, or (most generally) commutative ring (with unity) R replacing the integers. If R is a field, the cycle space is a vector space over R with dimension m - n + c, where c is the number of connected components of G. If R is any commutative ring, the cycle space is a free R-module with rank m - n + c.
When R is an abelian group such a cycle may also be called an R-flow on G. Nowhere-zero R-flows for a finite abelian group R of k elements are related to nowhere-zero integral k-flows in Tutte's theory. The number of nowhere-zero R-cycles is an evaluation of the Tutte polynomial, dual to the number of proper colorings of the graph (Tutte, 1984, Section IX.4).
Read more about this topic: Cycle Space
Famous quotes containing the words cycle, space, field and/or ring:
“Only mediocrities progress. An artist revolves in a cycle of masterpieces, the first of which is no less perfect than the last.”
—Oscar Wilde (1854–1900)
“Mere human beings can’t afford to be fanatical about anything.... Not even about justice or loyalty. The fanatic for justice ends by murdering a million helpless people to clear a space for his law-courts. If we are to survive on this planet, there must be compromises.”
—Storm Jameson (1891–1986)
“Love to chawnk green apples an’ go swimmin’ in the
lake.—
Hate to take the castor-ile they give for belly-ache!
‘Most all the time, the whole year round, there ain’t no flies on
me,
But jest ‘fore Christmas I’m as good as I kin be!”
—Eugene Field (1850–1895)
“When I received this [coronation] ring I solemnly bound myself in marriage to the realm; and it will be quite sufficient for the memorial of my name and for my glory, if, when I die, an inscription be engraved on a marble tomb, saying, “Here lieth Elizabeth, which reigned a virgin, and died a virgin.””
—Elizabeth I (1533–1603)