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:
“Oh, life is a glorious cycle of song,
A medley of extemporanea;
And love is a thing that can never go wrong;
And I am Marie of Roumania.”
—Dorothy Parker (1893–1967)
“Art and power will go on as they have done,—will make day out of night, time out of space, and space out of time.”
—Ralph Waldo Emerson (1803–1882)
“I see a girl dragged by the wrists
Across a dazzling field of snow,
And there is nothing in me that resists.
Once it would not be so....”
—Philip Larkin (1922–1986)
“Interpreting the dance: young women in white dancing in a ring can only be virgins; old women in black dancing in a ring can only be witches; but middle-aged women in colors, square dancing...?”
—Mason Cooley (b. 1927)