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علماء الرياضيات | Chromatic Invariant |
 1386
 02:16 صباحاً
date: 7-4-2022 |
Author : Biggs, N. L 
Book or Source : Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press 
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The chromatic invariant 
of a connected graph 
is the number of spanning trees of 
that have internal activity 1 and external activity 0.
For graphs other than the singleton graph (for which 
), it is also given by
 |
where 
is the vertex count and 
is the chromatic polynomial of 
.
A connected graph 
is separable iff 
and is a series-parallel graph iff 
(Biggs 1993, p. 109).
Unless otherwise stated, hydrogen atoms are usually ignored in the computation of such indices as organic chemists usually do when they write a benzene ring as a hexagon (Devillers and Balaban 1999, p. 25).
Special cases are summarized in the following table (Biggs 1993, p. 110).
| graph | OEIS |  ,  , ... |
| Andrásfai graph | A280333 | 1, 1, 12, 815, 157762, ... |
| antiprism graph | A294152 | X, X, 11, 38, 112, 309, 828, 2190, 5759, ... |
| Apollonian network | A000000 | 2, 16, 8192, ... |
 black bishop graph |
A295170 | 1, 1, 0, 8, 3528, 18475776, ... |
cocktail party graph  |
A295166 | 2, 1, 11, 362, 21234, 1965624, 264398280, ... |
complete bipartite graph  |
A048144 | 1, 1, 5, 73, 2069, 95401, 6487445, 610093513, ... |
complete tripartite graph  |
A182553 | 1, 11, 1243, 490043, 463370491, ... |
complete graph  |
A000142 | 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ... |
 -crossed prism graph |
A295168 | X, 11, 85, 521, 2869, 15017, 76717, 387425, ... |
| crown graph | A295171 | 1, 11, 328, 16369, 1181276, ... |
| cube-connected cycle graph | A000000 | X, X, 2816, ... |
empty graph  |
A000000 | 1, 2,  , 4,  , 6,  , 8,  , 10, ... |
| Fibonacci cube graph | A295927 | 1, 0, 0, 1, 36, 58432, ... |
| folded cube graph | A295172 | X, 1, 2, 73, 1872172, ... |
| gear graph | A000027 | X, X, 2, 3, 4, 5, 6, 7, 8, 9 ... |
grid graph  |
A182517 | 1, 1, 3, 72, ... |
grid graph  |
A295173 | 1, 11, 156284551, ... |
| halved cube graph | A295174 | X, 1, 1, 2, 362, 1784062800, ... |
| Hanoi graph | A295175 | 1, 64, 1073741824, ... |
hypercube graph  |
A295176 | 1, 1, 11, 48253, ... |
| Keller graph | A000000 | 4, 1872172, ... |
 king graph |
A295177 | 1, 2, 48, 31328, 473555616, ... |
 knight graph |
A295178 | 1, 4, -1, 78, 1306725, ... |
Möbius ladder  |
A000325 | X, X, 5, 12, 27, 58, 121, 248, 503, 1014, ... |
Mycielski graph  |
A000000 | 1, 1, 1, 238, ... |
odd graph  |
A000000 | 1, 1, 36, ... |
permutation star graph  |
A000000 | 1, 1, 1, 87837, ... |
prism graph  |
A000295 | X, X, 4, 11, 26, 57, 120, 247, 502, 1013, ... |
 queen graph |
A295187 | 1, 2, 1308, 1238775828, ... |
rook complement graph  |
A295186 | 1, 0, 98, 787211620, ... |
rook graph  |
A295184 | X, 1, 98, ... |
| Sierpiński sieve graph | A295189 | 1, 1, 27, 20346417, ... |
| Sierpiński tetrahedron graph | A000000 | 2, 176, ... |
| sun graph | A000142 | X, X, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... |
torus grid graph  |
A295191 | X, X, 98, 48253, 121790284, ... |
| transposition graph | A000000 | 1, 1, 5, ... |
| triangular graph | A295192 | X, 1, 1, 11, 3444, 32396796, ... |
| triangular grid graph | A295190 | 1, 1, 1, 5, 97, 6739, 1611097, 1295101469, ... |
| triangular honeycomb obtuse knight graph | A295194 | 1,  , 6, 0, 0, 11687, 100231463, ... |
| triangular honeycomb queen graph | A295195 | 1, 1, 11, 3714, 39103200, ... |
wheel graph  |
A000027 | X, X, X, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... |
 white bishop graph |
A295217 | 1, 1, 8, 2044, 18475776, ... |
Closed forms are summarized in the following table, where 
is a Lucas number, 
is a Stirling number of the second kind, and 
is a gamma function.
| graph | chromatic invariant formula |
| antiprism graph |  |
complete bipartite graph  |
![sum_(k=0)^(n-1)(k!)^2[S(n-1,k)]^2](https://mathworld.wolfram.com/images/equations/ChromaticInvariant/Inline46.svg) |
complete graph  |
 |
complete tripartite graph  |
 |
 -crossed prism graph |
 |
cycle graph  , |
1 |
empty graph  |
 |
| gear graph |  |
| helm graph | 0 |
ladder graph  |
1 |
ladder rung graph  |
 |
Möbius ladder  |
 |
| pan graph | 0 |
path graph  |
 |
prism graph  |
 |
star graph  |
 |
| sun graph |  |
sunlet graph |
0 |
| web graph | 0 |
wheel graph  |
 |
Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, pp. 107-109, 1993.
Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, 1999.
Sloane, N. J. A. Sequences A000027/M0472, A000142/M1675, A000295/M3416, A000325, and A048144 in "The On-Line Encyclopedia of Integer Sequences."
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