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)
“But alas! I never could keep a promise. I do not blame myself for this weakness, because the fault must lie in my physical organization. It is likely that such a very liberal amount of space was given to the organ which enables me to make promises, that the organ which should enable me to keep them was crowded out. But I grieve not. I like no half-way things. I had rather have one faculty nobly developed than two faculties of mere ordinary capacity.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“The woman ... turned her melancholy tone into a scolding one. She was not very young, and the wrinkles in her face were filled with drops of water which had fallen from her eyes, which, with the yellowness of her complexion, made a figure not unlike a field in the decline of the year, when the harvest is gathered in and a smart shower of rain has filled the furrows with water. Her voice was so shrill that they all jumped into the coach as fast as they could and drove from the door.”
—Sarah Fielding (1710–1768)
“I started out very quiet and I beat Turgenev. Then I trained hard and I beat de Maupassant. I’ve fought two draws with Stendhal, and I think I had an edge in the last one. But nobody’s going to get me in any ring with Tolstoy unless I’m crazy or I keep getting better.”
—Ernest Hemingway (1899–1961)