Stability Index

The stability index  of a graph  is defined by

where  is the th coefficient of the characteristic polynomial and  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 class	OEIS	, , ...
Andrásfai graph	A000000	2, 2, 70, 312, 2176, 16054, 69882, ...
antiprism graph	A000000	X, X, 13, 117, 606, 1453, 5013, 26141, 91867, ...
Apollonian network	A000000	10, 56, 11757, 108238346932, ...
cocktail party graph	A000000	1, 5, 13, 73, 281, 1101, 4005, 14225, 49201, ...
complete bipartite graph	A000000	2, 5, 10, 17, 26, 37, 50, 65, 82, 10, ...
complete graph	A001477	0, 2, 2, 10, 24, 66, 160, 386, 896, 2050, ...
complete tripartite graph	A000000	2, 13, 54, 49, 250, 109, ...
crossed prism graph	A000000	80, 1280, 11520, 128000, 1310720, 13844480, ...
crown graph	A000000	X, X, 20, 80, 272, 832, 2368, 6400, 16640, ...
cube-connected cycle graph	A000000	X, X, 702672, 2312110080000000, ...
cycle graph	A000000	X, X, 2, 5, 2, 20, 2, 45, 2, 125, 2, 320, 2, ...
empty graph	A000000	0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
folded cube graph	A000000	2, 10, 17, 150806, 1129150390625, ...
grid graph	A000000	0, 5, 0, 4176, 0, 390590941, 0, 4363974545574685, ...
grid graph	A000000	0, 80, 0, 473138263916015625, 0, ...
halved cube graph	A000000	0, 2, 10, 73, 1716177, 990952398727038, ...
hypercube graph	A000000	2, 5, 80, 10625, 2662400000, 27254945465087890625, ...
Möbius ladder	A000000	X, X, 10, 70, 250, 580, 1690, 4870, 13690, 40650, 118810, ...
Mycielski graph	A000000	0, 2, 2, 432, 2440488, 113920311278592, ...
odd graph	A000000	0, 2, 424, 46757736448, ...
pan graph	A000000	X, X, 6, 0, 16, 0, 42, 0, 110, 0, ...
path graph	A000000	0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233, ...
permutation star graph	A000000	0, 2, 20, 1250000, ...
prism graph	A000000	X, X, 22, 80, 204, 500, 1684, 5120, 14062, ...
rook graph	A000000	5, 112, 492593, 77149243008, 254773522981613013, ...
star graph	A000290	0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...
sun graph	A000000	X, X, 23, 52, 115, 202, 761, 4390, 20771, ...
sunlet graph	A000000	X, X, 14, 32, 82, 200, 478, 1152, 2786, 6728, ...
tetrahedral graph	A000000	X, X, X, X, X, 88567324, 6545276817256, ...
triangular graph	A000000	X, 0, 2, 13, 748, 362144, 2185830840, ...
web graph	A000000	X, X, 333, 56, 6253, 566, 120213, 2982, 2202832, ...
wheel graph	A000000	X, X, X, 10, 8, 26, 48, 78, 144, 220, 500, 650, ...

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

graph	stability index
cocktail party graph
complete bipartite graph
complete graph
crossed prism graph
crown graph
cycle graph
empty graph
gear graph	0
pan graph
path graph
star graph
sunlet graph

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.