The reliability polynomial
As we all know, the topological structure of a realworld complex system is often described by a graph. And recently, researchers tend to represent some more complex systems by hypergraphs [
One of the most common measures in network reliability is the allterminal reliability [
As a generalization of a graph, the hypergraph provides an effective method for network description [
Here we define the probability as
In [
Trees are among the most fundamental, useful, and understandable objects in all of graph theory. This kind of common sense is also true for hypergraphs [
The network reliability with the edge failure is closely related to the number of spanning trees of the corresponding graph [
The remaining parts of this paper are organized as follows. Firstly, we introduce some necessary definitions and notations. Then we present recursive relations for the reliability polynomial with edge failure of a
Many definitions of hypergraphs would be naturally extended from graphs; undefined terms can be found in [
In the following parts, we conduct the same researches about reliability with edge failure of graphs on hypergraphs. Let
The reliability polynomial of a hypergraph is often presented in the literature as
The tree is an important object in graph theory, and the number of spanning trees of a graph is closely related to the reliability of this graph [
With regard to the lower bound of the number of edges of connected
Let
In this section, we present a recurrence relation for the computation of the reliability polynomial of
The reliability polynomial of the complete graph
Now we consider the reliability of a
The reliability polynomial of the
Let us fix a vertex
In Theorem
Table
Reliability polynomial for small hypergraphs.
r  n  

4  5  6  7  
3 






4 






5 





6 

When
The reliability polynomials of the complete hypergraph
In the following, we give the equivalent operation of the recurrence formula, then we can get some properties of the corresponding
We now give an example of transformation of the reliability polynomial
From these equations, we find that
According to the definition of the hypertree in this paper and the theory of network reliability, the spanning hypertree of
Combined with the above recurrence relation of the
The number of spanning hypertrees in
Let
Based on Cases 1 and 2, the conclusions we need to prove have been established.
Table
The number of spanning trees for small hypergraphs.
r  n  

4  5  6  7  8  
3 

15  480 

117810 


4 



14560  


5 


280  


6 

210  


7  28 
According to the definition of the spanning hypertree above and standard expression of reliability in
About enumeration of spanning hypertrees in
If
Let
On the other hand, the edge number of spanning hypertrees in
So far, we have completed the proof of Theorem
By Theorem
As a corollary to Theorem
Let
The transformation of the recurrence in Theorem
In a similar fashion as stated in Theorem
Let
Because
In this paper, a recursive algorithm is proposed for calculating the allterminal reliability of a
There still exist some interesting open questions for future research in this field:
How can we get a recurrence relation for the
How can we calculate 2edgeconnected (or higher) reliability of
How can we calculate more about the result of the number of spanning hypertrees in
For a
How can we attack the problem for general hypergraphs?
No data were used to support this study.
The authors declare that they have no conflicts of interest.
This work is supported by NSFC (Grants nos. 11661069, 61663041, and 61763041), the Science Found of Qinghai Province (Grants nos. 2015ZJ723 and 2018ZJ718), and the Fundamental Research Funds for the Central Universities (Grant no. 2017TS045).