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 lifelong process of caregiving, is the ultimate link between caregivers of all ages. You and I are not just in a phase we will outgrow. This is life—birth, death, and everything in between.... The care continuum is the cycle of life turning full circle in each of our lives. And what we learn when we spoon-feed our babies will echo in our ears as we feed our parents. The point is not to be done. The point is to be ready to do again.”
—Paula C. Lowe (20th century)
“If we remembered everything, we should on most occasions be as ill off as if we remembered nothing. It would take us as long to recall a space of time as it took the original time to elapse, and we should never get ahead with our thinking. All recollected times undergo, accordingly, what M. Ribot calls foreshortening; and this foreshortening is due to the omission of an enormous number of facts which filled them.”
—William James (1842–1910)
“Something told the wild geese
It was time to go.
Though the fields lay golden
Something whispered—”Snow.””
—Rachel Lyman Field (1894–1942)
“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)