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 Buddha, the Godhead, resides quite as comfortably in the circuits of a digital computer or the gears of a cycle transmission as he does at the top of a mountain or in the petals of a flower.”
—Robert M. Pirsig (b. 1928)
“Let the space under the first storey be dark, let the water
lap the stone posts, and vivid green slime glimmer
upon them; let a boat be kept there.”
—Denise Levertov (b. 1923)
“the whole field is a
white desire, empty, a single stem;
a cluster, flower by flower,
a pious wish to whiteness gone over—
or nothing.”
—William Carlos Williams (1883–1963)
“Roll unmanly over this turning tuft,
O ring of seas, nor sorrow as I shift
From all my mortal lovers with a starboard smile....”
—Dylan Thomas (1914–1953)