Let be a graph, and suppose each edge of is independently deleted with fixed probability . Then the probability that no connected component of is disconnected as a result, denoted is known as the reliability polynomial of .
The reliability polynomial is directly expressible in terms of the Tutte polynomial of a given graph as
(1)
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where is the vertex count, the edge count, and the number of connected components (Godsil and Royle 2001, p. 358; error corrected). This is equivalent to the definition
(2)
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where is the number of subgraphs of the original graph having exactly edges and for which every pair of nodes in is joined by a path of edges lying in subgraph (i.e., is connected and ), which is the definition due to Page and Perry (1994) after making the change .
For example, the reliability polynomial of the Petersen graph is given by
(3)
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(Godsil and Royle 2001, p. 355).
The following table summarizes simple classes of graphs having closed-form reliability polynomials. Here, .
The following table summarizes the recurrence relations for reliability polynomials for some simple classes of graphs.
graph | order | recurrence |
cycle graph | 2 | |
gear graph | 3 | |
ladder graph | 2 | |
pan graph | 2 | |
path graph | 1 | |
star graph | 1 | |
sunlet graph | 2 | |
wheel graph | 3 |
Nonisomorphic graphs do not necessarily have distinct reliability polynomials. The following table summarizes some co-reliability graphs.
reliability polynomial | graphs | |
4 | claw graph, path graph | |
5 | fork graph, path graph , star graph | |
5 | bull graph, cricket graph, -tadpole graph | |
5 | dart graph, kite graph |