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)
“It is not through space that I must seek my dignity, but through the management of my thought. I shall have no more if I possess worlds.”
—Blaise Pascal (1623–1662)
“... there are no chains so galling as the chains of ignorance—no fetters so binding as those that bind the soul, and exclude it from the vast field of useful and scientific knowledge. O, had I received the advantages of early education, my ideas would, ere now, have expanded far and wide; but, alas! I possess nothing but moral capability—no teachings but the teachings of the Holy Spirit.”
—Maria Stewart (1803–1879)
“With this Ring I thee wed, with my body I thee worship, and with all my worldly goods I thee endow.”
—Book Of Common Prayer, The. Solemnization of Matrimony, “Wedding,” (1662)