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)
“Though seas and land be ‘twixt us both,
Our faith and troth,
Like separated souls,
All time and space controls:
Above the highest sphere we meet
Unseen, unknown, and greet as angels greet.”
—Richard Lovelace (1618–1658)
“But the old world was restored and we returned
To the dreary field and workshop, and the immemorial feud
Of rich and poor. Our victory was our defeat.”
—Sir Herbert Read (1893–1968)
“It is engend’red in the eyes,
With gazing fed; and fancy dies
In the cradle where it lies.
Let us all ring fancy’s knell.
I’ll begin it—Ding, dong, bell.”
—William Shakespeare (1564–1616)