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:
“The cycle of the machine is now coming to an end. Man has learned much in the hard discipline and the shrewd, unflinching grasp of practical possibilities that the machine has provided in the last three centuries: but we can no more continue to live in the world of the machine than we could live successfully on the barren surface of the moon.”
—Lewis Mumford (1895–1990)
“Here in the U.S., culture is not that delicious panacea which we Europeans consume in a sacramental mental space and which has its own special columns in the newspapers—and in people’s minds. Culture is space, speed, cinema, technology. This culture is authentic, if anything can be said to be authentic.”
—Jean Baudrillard (b. 1929)
“He stung me first and stung me afterward.
He rolled me off the field head over heels
And would not listen to my explanations.”
—Robert Frost (1874–1963)
“I was exceedingly interested by this phenomenon, and already felt paid for my journey. It could hardly have thrilled me more if it had taken the form of letters, or of the human face. If I had met with this ring of light while groping in this forest alone, away from any fire, I should have been still more surprised. I little thought that there was such a light shining in the darkness of the wilderness for me.”
—Henry David Thoreau (1817–1862)