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Stability Index

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The stability index Z^_(G) of a graph G is defined by

Z^_=sum_(k=0)^(|_n/2_|)|c_(2k)|,

where c_k is the kth coefficient of the characteristic polynomial and |_n_| denotes the floor function.

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).

The following table summarizes values of the Kirchhoff sum index for various special classes of graphs.

graph classOEISZ^_(G_1), Z^_(G_2), ...
Andrásfai graphA0000002, 2, 70, 312, 2176, 16054, 69882, ...
antiprism graphA000000X, X, 13, 117, 606, 1453, 5013, 26141, 91867, ...
Apollonian networkA00000010, 56, 11757, 108238346932, ...
cocktail party graph K_(n×2)A0000001, 5, 13, 73, 281, 1101, 4005, 14225, 49201, ...
complete bipartite graph K_(n,n)A0000002, 5, 10, 17, 26, 37, 50, 65, 82, 10, ...
complete graph K_nA0014770, 2, 2, 10, 24, 66, 160, 386, 896, 2050, ...
complete tripartite graph K_(n,n,n)A0000002, 13, 54, 49, 250, 109, ...
crossed prism graphA00000080, 1280, 11520, 128000, 1310720, 13844480, ...
crown graphA000000X, X, 20, 80, 272, 832, 2368, 6400, 16640, ...
cube-connected cycle graphA000000X, X, 702672, 2312110080000000, ...
cycle graph C_nA000000X, X, 2, 5, 2, 20, 2, 45, 2, 125, 2, 320, 2, ...
empty graph K^__nA0000000, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
folded cube graphA0000002, 10, 17, 150806, 1129150390625, ...
grid graph P_n square P_nA0000000, 5, 0, 4176, 0, 390590941, 0, 4363974545574685, ...
grid graph P_n square P_n square P_nA0000000, 80, 0, 473138263916015625, 0, ...
halved cube graphA0000000, 2, 10, 73, 1716177, 990952398727038, ...
hypercube graph Q_nA0000002, 5, 80, 10625, 2662400000, 27254945465087890625, ...
Möbius ladder M_nA000000X, X, 10, 70, 250, 580, 1690, 4870, 13690, 40650, 118810, ...
Mycielski graphA0000000, 2, 2, 432, 2440488, 113920311278592, ...
odd graph O_nA0000000, 2, 424, 46757736448, ...
pan graphA000000X, X, 6, 0, 16, 0, 42, 0, 110, 0, ...
path graphA0000000, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233, ...
permutation star graph PS_nA0000000, 2, 20, 1250000, ...
prism graph Y_nA000000X, X, 22, 80, 204, 500, 1684, 5120, 14062, ...
rook graph K_n square K_nA0000005, 112, 492593, 77149243008, 254773522981613013, ...
star graph S_nA0002900, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...
sun graphA000000X, X, 23, 52, 115, 202, 761, 4390, 20771, ...
sunlet graph C_n circledot K_1A000000X, X, 14, 32, 82, 200, 478, 1152, 2786, 6728, ...
tetrahedral Johnson graphA000000X, X, X, X, X, 88567324, 6545276817256, ...
triangular graphA000000X, 0, 2, 13, 748, 362144, 2185830840, ...
web graphA000000X, X, 333, 56, 6253, 566, 120213, 2982, 2202832, ...
wheel graph W_nA000000X, X, X, 10, 8, 26, 48, 78, 144, 220, 500, 650, ...

Closed forms are summarized in the following table, where F_n is a Fibonacci number.

graphstability index
cocktail party graph K_(n×2)1/6(3^n(2n-3)+3(-1)^n(2n-1)+12)
complete bipartite graph K_(n,n)n^2+1
complete graph K_n2^(n-2)(n-2)+(-1)^n+1
crossed prism graph(6+2sqrt(5))^n+(6-2sqrt(5))^n-(-1)^n2^(2n+1)
crown graph2^(n-1)(n^2-2n+2)
cycle graph C_n{2 for n odd; 5F_(n/2)^2 for n even
empty graph K^__n1/2(1+(-1)^n)
gear graph0
pan graph{0 for n even; 2F_(n+1) for n odd
path graph P_n{0 for n odd; F_(n+1) for n even
star graph S_n1/2(1+(-1)^n)n
sunlet graph C_n circledot K_1(1-sqrt(2))^n+(1+sqrt(2))^n-2cos(1/2npi)

See also

Matching Polynomial, Hosoya Index

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References

Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 27-28 and 105, 1999.

Referenced on Wolfram|Alpha

Stability Index

Cite this as:

Weisstein, Eric W. "Stability Index." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StabilityIndex.html

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