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)
“It is the space inside that gives the drum its sound.”
—Hawaiian saying no. 1189, ‘lelo No’Eau, collected, translated, and annotated by Mary Kawena Pukui, Bishop Museum Press, Hawaii (1983)
“The frequent failure of men to cultivate their capacity for listening has a profound impact on their capacity for parenting, for it is mothers more than fathers who are most likely to still their own voices so they may hear and draw out the voices of their children.”
—Mary Field Belenky (20th century)
“What is a novel? I say: an invented story. At the same time a story which, though invented has the power to ring true. True to what? True to life as the reader knows life to be or, it may be, feels life to be. And I mean the adult, the grown-up reader. Such a reader has outgrown fairy tales, and we do not want the fantastic and the impossible. So I say to you that a novel must stand up to the adult tests of reality.”
—Elizabeth Bowen (1899–1973)