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)
“When Paul Bunyan’s loggers roofed an Oregon bunkhouse with shakes, fog was so thick that they shingled forty feet into space before discovering they had passed the last rafter.”
—State of Oregon, U.S. public relief program (1935-1943)
“Last night I watched my brothers play,
The gentle and the reckless one,
In a field two yards away.
For half a century they were gone
Beyond the other side of care
To be among the peaceful dead.”
—Edwin Muir (1887–1959)
“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)