Sigma Polynomial
المؤلف:
Frucht, R. W. and Giudici, R. E
المصدر:
"Some Chromatically Unique Graphs with Seven Points." Ars Combin. A 16,
الجزء والصفحة:
...
29-4-2022
1723
Sigma Polynomial
Let a simple graph 
have 
vertices, chromatic polynomial 
, and chromatic number 
. Then 
can be written as
where 
and 
is a falling factorial, and the polynomial
is known as the 
-polynomial (Frucht and Giudici 1983; Li et al. 1987; Read and Wilson 1998, p. 265).
-polynomials for a number of simple graphs are summarized in the following table.
graph  |
 |
claw graph  |
 |
complete graph  |
1 |
| cubical graph |
 |
cycle graph  |
 |
| octahedral graph |
 |
path graph  |
 |
pentatope graph  |
1 |
square graph  |
 |
star graph  |
 |
star graph  |
 |
tetrahedral graph  |
1 |
triangle graph  |
1 |
wheel graph  |
 |
wheel graph  |
 |
REFERENCES
Frucht, R. W. and Giudici, R. E. "Some Chromatically Unique Graphs with Seven Points." Ars Combin. A 16, 161-172, 1983.
Korfhage, R. R. "
-Polynomials and Graph Coloring." J. Combin. Th. Ser. B 24, 137-153, 1978.
Li, N.-Z.; Whitehead, E. G. Jr.; and Xu, S.-J. "Classification of Chromatically Unique Graphs Having Quadratic 
-Polynomials." J. Graph Th. 11, 169-176, 1987.
Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 265, 1998.
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