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)
“And Space with gaunt grey eyes and her brother Time
Wheeling and whispering come,”
—James Elroy Flecker (1884–1919)
“I don’t like comparisons with football. Baseball is an entirely different game. You can watch a tight, well-played football game, but it isn’t exciting if half the stadium is empty. The violence on the field must bounce off a lot of people. But you can go to a ball park on a quiet Tuesday afternoon with only a few thousand people in the place and thoroughly enjoy a one-sided game. Baseball has an aesthetic, intellectual appeal found in no other team sport.”
—Bowie Kuhn (b. 1926)
“Ring out, ring out thy mournful rhymes,
But ring the fuller minstrel in.”
—Alfred Tennyson (1809–1892)