Research ArticleA Recursive Formula for the Reliability of a 𝑟-Uniform CompleteHypergraph and Its Applications

Ke Zhang ,1,2,3 Haixing Zhao ,1,2,3,4 Zhonglin Ye ,2,3,4 and Lixin Dong1,2,3

1 School of Computer, Qinghai Normal University, Xining 810008, China2Key Laboratory of Tibetan Information Processing and Machine Translation in QH, Xining 810008, China3Key Laboratory of Tibetan Information Processing, Ministry of Education, Xining 810008, China4School of Computer Science, Shaanxi Normal University, Xi’an 710062, China

Correspondence should be addressed to Haixing Zhao; h.x.zhao@163.com

Received 25 August 2018; Accepted 24 September 2018; Published 14 October 2018

Academic Editor: Frederico R. B. Cruz

Copyright © 2018 Ke Zhang et al.This is an open access article distributed under theCreative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The reliability polynomial 𝑅(𝑆, 𝑝) of a finite graph or hypergraph 𝑆 = (𝑉, 𝐸) gives the probability that the operational edges orhyperedges of 𝑆 induce a connected spanning subgraph or subhypergraph, respectively, assuming that all (hyper)edges of 𝑆 failindependentlywith an identical probability𝑞 = 1−𝑝. In this paper,we investigate the probability that the hyperedges of a hypergraphwith randomly failing hyperedges induce a connected spanning subhypergraph.The computation of the reliability for (hyper)graphsis an NP-hard problem. We provide recurrence relations for the reliability of 𝑟-uniform complete hypergraphs with hyperedgefailure. Consequently, we determine and calculate the number of connected spanning subhypergraphs with given size in the 𝑟-uniform complete hypergraphs.

1. Introduction

As we all know, the topological structure of a real-worldcomplex system is often described by a graph. And recently,researchers tend to represent some more complex systems byhypergraphs [1–4]. With the establishment and perfection ofthe hypergraph theory by Berge et al. [5, 6], many interestingobjects in graphs were extended to hypergraphs in a naturalway. Since a graph is a 2-uniform hypergraph, many of thecorresponding results about graphs could be analogized tohypergraphs, and there exist some representative works. Forexample, Dankelmann et al. [7] and Zhao et al. [8] obtainedseveral sufficient conditions for hypergraphs to bemaximallyedge-connected. Gu et al. characterized the degree sequenceof a uniform hypergraph, and this degree sequence makesthe uniform hypergraph 𝑘-edge-connected [9]. In [10] and[11], an upper bound of the sum of squares of degrees in agraph and a hypergraph was given, respectively. Moreover,hypergraph theory can be applied to optimize wireless com-munication networks [12]. The network reliability analysisand synthesis have attracted much attention and there exist

various results in [13–16], but in the same field there existseldom results for hypergraphs.

One of the most common measures in network relia-bility is the all-terminal reliability [14]. Suppose that 𝐺 =(𝑉, 𝐸) is a graph where the edges independently fail withthe same probability 𝑞. The all-terminal reliability is theprobability that the surviving edges induce a connectedspanning subgraph. It can be expressed as a polynomial𝑅(𝐺, 𝑞) = ∑𝑒𝑖=𝑛−1 𝑠𝑖(𝐺)(1 − 𝑞)𝑖𝑞𝑒−𝑖, where 𝑠𝑖(𝐺) is the numberof connected spanning subgraphs of 𝐺 containing 𝑖 edges.The problems of calculating the all-terminal reliability anddesigning reliable network have been probed and studieddeeply in [14].

As a generalization of a graph, the hypergraph provides aneffective method for network description [1, 2, 17]. A networkrepresented by a hypergraph is called a hypernetwork. Moreand more researchers are interested in hypergraphs or hyper-networks. The existing literatures about hypernetworks oftenfocus on the establishment of models [18–20] or topologicalproperties [1, 21, 22]. Extensive researches have been con-ducted in the field of edge-connectivity of hypergraphs [7–9].

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 3131087, 7 pageshttps://doi.org/10.1155/2018/3131087

2 Mathematical Problems in Engineering

In this paper, we investigate the hypernetwork reliability withedge failure.

Here we define the probability as 𝑅(𝐻, 𝑝) that theoperational edges of the hypergraph 𝐻 induce a connectedspanning subhypergraph. Our main results are recurrencerelations for the calculation of 𝑅(𝐾𝑟𝑛, 𝑝), where 𝐾𝑟𝑛 denotesthe 𝑟-uniform complete hypergraphs of order 𝑛. Conse-quently, we provide a new method to calculate the num-ber of connected spanning subhypergraphs in 𝐾𝑟𝑛. The 𝑟-uniform complete hypergraphs have long been in the focus ofhypernetwork reliability research for different reasons. Theyappear as subhypergraphs in larger hypernetworks usually.They are the densest ones among all 𝑟-uniform hypergraphs,so if we obtain the reliability of the 𝑟-uniform completehypergraphs, the upper bounds of the reliability of the 𝑟-uniform hypergraphs can be found satisfactory. And theyhave a nice combinatorial structure that allows researcher toexplore their symmetry. For some applications, see [23–25].

In [26] Gilbert presented a recursive algorithm for thecalculation of the all-terminal reliability of a complete graph𝐾𝑛 by fixing a vertex V ∈ 𝑉 and considered the relationshipsbetween all connected subgraphs containing V of order 𝑘 andcorresponding subgraphs of order 𝑛 − 𝑘, then the probability𝑅(𝐾𝑛, 𝑝) is exactly equivalent to the probability that theinduced spanning subgraph is connected for 𝑛 vertices set.We substantially generalize this idea and propose the result inhypergraphs based on above analogies, and the research resultfrom Gilbert is a special case of the research achievementsproposed by this paper.

Trees are among the most fundamental, useful, andunderstandable objects in all of graph theory. This kind ofcommon sense is also true for hypergraphs [27–29]. Anotherimportant part of this paper is to generalize the notion ofa tree to uniform hypergraphs and to investigate enumera-tion of connected spanning subhypergraphs, as to calculatethe number of spanning hypertrees in 𝑟-uniform completehypergraphs.

The network reliability with the edge failure is closelyrelated to the number of spanning trees of the correspondinggraph [30]. We obtain the number of spanning hypertrees in𝑟-uniform complete hypergraphs with the inspiration of sucha similar relationship in general graphs.

The remaining parts of this paper are organized as follows.Firstly, we introduce some necessary definitions and nota-tions. Then we present recursive relations for the reliabilitypolynomial with edge failure of a 𝑟-uniform complete hyper-graph. Counting spanning hypertree in a 𝑟-uniform completehypergraph is researched in Section 3. Consequently, wefinish the entire paper with some conclusions and openproblems in Section 4.

1.1. Nomenclature andDefinitions. Manydefinitions of hyper-graphs would be naturally extended from graphs; undefinedterms can be found in [5, 6]. A hypergraph𝐻 is a pair (𝑉, 𝜀),where 𝑉 is the vertex set of𝐻 and 𝜀 is a collection of distinctnonempty subsets of 𝑉. An element in 𝜀 is a hyperedge orsimply an edge of 𝐻. We consider the hypergraph with noisolated vertices, which are not contained in any edge. Let𝜀 = {𝐸1, 𝐸2, ⋅ ⋅ ⋅ , 𝐸𝑚}. A hypergraph 𝐻 is simple hypergraph

if 𝐸𝑖 ⊆ 𝐸𝑗 implies that 𝑖 = 𝑗 for any 𝑖, 𝑗 with 1 ≤ 𝑖, 𝑗 ≤ 𝑚. Thehypergraphs researched in this paper are simple hypergraphs.A hypergraph𝐻 is a 𝑟-uniform hypergraph if |𝐸𝑖| = 𝑟 for each𝑖 with 1 ≤ 𝑖 ≤ 𝑚. Thus, a graph is a 2-uniform hypergraph,and vice versa. Let 𝑛, 𝑟 be integer and hold 2 ≤ 𝑟 ≤ 𝑛 − 1.Wedefine the 𝑟-uniform complete hypergraph of order 𝑛 as ahypergraph denoted as 𝐾𝑟𝑛 containing all the 𝑟-subsets of theset 𝑉 of cardinality 𝑛.

In the following parts, we conduct the same researchesabout reliability with edge failure of graphs on hypergraphs.Let V denote an arbitrary vertex in 𝐻. We assume that alledges of 𝐻 fail independently with identical probability 𝑞.The surviving subhypergraph of 𝐻 is called the connectedspanning subhypergraph of 𝐻. The reliability polynomial𝑅(𝐻, 𝑞) gives the probability of the surviving subhypergraphof a hypergraph𝐻(𝑉, 𝜀).

The reliability polynomial of a hypergraph is often pre-sented in the literature as 𝑅(𝐻, 𝑝) with 𝑝 = 1 − 𝑞. We write𝑅(𝐻, 𝑞) for simpler presentation as a function of the edgefailure probability 𝑞.

The tree is an important object in graph theory, andthe number of spanning trees of a graph is closely relatedto the reliability of this graph [30]. The definition of a treecould be extended to the hypergraph, which is consequentlycalled a hypertree or simply a tree. Although there aremany statements of the definition of a tree in graph theory,these statements are equivalent and intuitive. However, thegeneralization of the definition of a tree in hypergraphs ismuchmore complicated, and descriptions of these definitionsare completely different [27–29]. In this paper, a hypertreeis defined as a connected hypergraph which is disconnectedby removing any edge among the edge set [27]. A spanninghypertree of 𝐻 is a spanning subhypergraph with the mini-mum number of edges of 𝐻 that is a hypertree. We denotethe number of spanning trees of𝐻 by 𝜏(𝐻).

With regard to the lower bound of the number of edgesof connected 𝑟-uniform hypergraphs, Lai et al. [9] give thefollowing results.

Theorem 1. Let𝐻 be a 𝑟-uniform hypergraph with 𝑛 vertexes.If 𝐻 is connected, then 𝑚 ≥ (𝑛 − 1)/(𝑟 − 1). Moreover, theequality holds if and only if 𝐻 − 𝐸 has 𝑟 components for anyedge 𝐸 which belongs to 𝜀(𝐻).2. A Recursive Formula for the Reliability of a𝑟-Uniform Complete Hypergraph

In this section, we present a recurrence relation for the com-putation of the reliability polynomial of 𝑟-uniform completehypergraphs. As a preparation, we restate a classical theoremconcerning the all-terminal reliability of a complete graph. Bydistinguishing one vertex of a complete graph, Gilbert [26]obtained the following results.

Theorem 2. The reliability polynomial of the complete graph𝐾𝑛(𝑛 ≥ 2) satisfies the following recurrence relations:𝑅 (𝐾𝑛, 𝑞) = 1 − 𝑛−1∑

𝑘=1(𝑛 − 1𝑘 − 1)𝑅 (𝐾𝑘, 𝑞) 𝑞𝑘(𝑛−𝑘). (1)

Mathematical Problems in Engineering 3

Table 1: Reliability polynomial for small hypergraphs.

r n4 5 6 7

3 1 − 4𝑞3 + 3𝑞4 1 − 5𝑞6 + 10𝑞9 − 6𝑞10 1 − 6𝑞10 + 15𝑞16 − 10𝑞18 1 − 7𝑞15 + 21𝑞25 − 35𝑞30 + 140𝑞33 − 210𝑞34+90𝑞354 1 − 5𝑞4 + 4𝑞5 1 − 6𝑞10 + 15𝑞14 − 10𝑞15 1 − 7𝑞20 + 21𝑞30 − 35𝑞34 + 20𝑞355 1 − 6𝑞5 + 5𝑞6 1 − 7𝑞15 + 21𝑞20 − 15𝑞216 1 − 7𝑞6 + 6𝑞7

Equation (1) and the initial condition 𝑅(𝐾1, 𝑞) = 1 uniquelydetermine the value 𝑅(𝐾𝑛, 𝑞).2.1.TheReliability of a 𝑟-UniformComplete Hypergraph. Nowwe consider the reliability of a 𝑟-uniform complete hyper-graph. In view of Theorem 2 we investigate the reliabilitypolynomial of the 𝑟-uniform complete hypergraph. Ourmainresult below generalizes Gilbert’s result.

Theorem 3. The reliability polynomial of the 𝑟-uniform com-plete hypergraph 𝐾𝑟𝑛, where 𝑛 ≥ 2, 2 ≤ 𝑟 ≤ 𝑛 − 1, satisfies thefollowing recurrence relations:

𝑅 (𝐾𝑟𝑛, 𝑞) = 1 − (𝑛 − 10 )𝑅 (𝐾1, 𝑞) 𝑞( 𝑛−1𝑟−1 )− 𝑛−1∑𝑘=𝑟

(𝑛 − 1𝑘 − 1)𝑅 (𝐾𝑟𝑘, 𝑞) 𝑞( 𝑛𝑟 )−(𝑘𝑟 )−( 𝑛−𝑘𝑟 ).(2)

Equation (2) uniquely determines 𝑅(𝐾𝑟𝑛, 𝑞) based on the initialvalues𝑅(𝐾1, 𝑞) = 1, 𝑅(𝐾𝑘𝑘 , 𝑞) = 1−𝑞, and the condition ( 𝑛𝑚 ) =0 (𝑛 < 𝑚).Proof. Let us fix a vertex V ∈ 𝑉(𝐾𝑟𝑛). The proof process iscompleted by the following two steps.

Step 1. The first quantity on the right side of the equationthat is subtracted from 1 is equal to 𝑞( 𝑛−1𝑟−1 ), which gives theprobability that makes the vertex V isolated from the 𝐾𝑟𝑛.There are ( 𝑛−1𝑟−1 ) edges in 𝐾𝑟𝑛 containing V.

Step 2. The second quantity is the form of summation as∑𝑛−1𝑘=𝑟 ( 𝑛−1𝑘−1 ) 𝑅(𝐾𝑟𝑘, 𝑞)𝑞(𝑛𝑟 )−(𝑘𝑟 )−(𝑛−𝑘𝑟 ), which gives the proba-bility that the root vertex V exactly builds the connectedcomponent with 𝑘 − 1 vertices, where all such connectedcomponents of order 𝑘must contain vertex V.There are ( 𝑛−1𝑘−1 )possible items to select the vertex set 𝐴 of order 𝑘 − 1 in theremaining 𝑛− 1 vertices. Then 𝑅(𝐾𝑟𝑘, 𝑞) is the probability thatthe set 𝐵 = 𝐴 ∪ {V} induces a connected subhypergraph.In the special case 𝑟 = 𝑘, the hypergraph with only oneedge containing all 𝑘 vertices is denoted as 𝐾𝑘𝑘 . According tothe definition of the reliability of the hypernetwork with theedge failure, we can obtain 𝑅(𝐾𝑘𝑘 , 𝑞) = 1 − 𝑞. The expression𝑞( 𝑛𝑟 )−(𝑘𝑟 )−(𝑛−𝑘𝑟 ) is the probability that all edges fail between𝐾𝑟𝑛 − 𝐵 and 𝐵 in 𝑟-uniform complete hypergraph 𝐾𝑟𝑛. Finally,we adopt the expression ( 𝑛𝑟 ) − ( 𝑘𝑟 ) − ( 𝑛−𝑘𝑟 ) to calculate the

number of edges connecting a component of order 𝑛 − 𝑘 andthe other induced by 𝐵.

In Theorem 3, the special case 𝑟 = 2 is the Gilbert’s resultinTheorem 2.

Table 1 shows some polynomials for the reliability of 𝑟-uniform complete hypergraphs 𝐾𝑟𝑛 of orders 4, 5, 6, and 7.

When 𝑟 approximates 𝑛, the terms in the recurrence forcalculating 𝑅(𝐾𝑟𝑛, 𝑞) are finite inTheorem 3, so we can get theexpression of 𝑅(𝐾𝑟𝑛, 𝑞) about 𝑛.The following Corollary 4 canbe yielded by setting 𝑟 = 𝑛 − 1, 𝑛 − 2, and 𝑛 − 3.Corollary4. Thereliability polynomials of the complete hyper-graph 𝐾𝑟𝑛 with 𝑟 = 𝑛 − 1, 𝑛 − 2, and 𝑛 − 3 are given by thefollowing:

𝑅 (𝐾𝑛−1𝑛 , 𝑞) = 1 − 𝑛𝑞𝑛−1 + (𝑛 − 1) 𝑞𝑛 (𝑛 ≥ 3) ;𝑅 (𝐾𝑛−2𝑛 , 𝑞) = 1 − 𝑛𝑞( 𝑛−12 ) + (𝑛2) 𝑞( 𝑛2 )−1

− (𝑛 − 12 ) 𝑞( 𝑛2 ) (𝑛 ≥ 4) ;𝑅 (𝐾𝑛−3𝑛 , 𝑞) = 1 − 𝑛𝑞( 𝑛−13 ) + (𝑛2) 𝑞( 𝑛3 )−𝑛+2

− (𝑛3) 𝑞( 𝑛3 )−1 + (𝑛 − 13 ) 𝑞( 𝑛3 )(𝑛 ≥ 5) .

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3. Enumeration of Spanning Hypertrees in𝑟-Uniform Complete Hypergraphs

3.1. Standard Version from the Recurrence. In the following,we give the equivalent operation of the recurrence formula,then we can get some properties of the corresponding 𝑟-uniform complete hypergraph. For example, we investigatethe number of connected spanning subhypergraphs inducedby the operational edges of a 𝑟-uniform complete hypergraphwith random failure of edges. In particular, the number ofspanning trees of a complete hypergraph can be obtaineddirectly. In a similar way proposed by the reliability of graph,the all-terminal reliability of a hypergraph can be defined asan equivalent form, which is the homogeneous polynomial

4 Mathematical Problems in Engineering

of 1 − 𝑞 and 𝑞. For a 𝑟-uniform complete hypergraph, thisstandard form is as follows:

𝑅 (𝐾𝑟𝑛, 𝑞) = ( 𝑛𝑟 )∑𝑖=0

𝑠𝑖 (𝐾𝑟𝑛) (1 − 𝑞)𝑖 𝑞( 𝑛𝑟 )−𝑖, (4)

where 𝑠𝑖(𝐾𝑟𝑛) denotes the number of connected spanningsubhypergraphs containing 𝑖 edges in a 𝑟-uniform completehypergraph.

We now give an example of transformation of the reliabil-ity polynomial 𝑅(𝐾36 , 𝑞) as follows.Example 5.

𝑅 (𝐾36 , 𝑞) = 1 − 6𝑞10 + 15𝑞16 − 10𝑞18= 20∑𝑖=0(20𝑖 ) (1 − 𝑞)𝑖 𝑞20−𝑖

− 6[ 10∑𝑖=0(10𝑖 ) (1 − 𝑞)𝑖 𝑞10−𝑖]𝑞10

+ 15[ 4∑𝑖=0(4𝑖) (1 − 𝑞)𝑖 𝑞4−𝑖] 𝑞16

− 10[ 2∑𝑖=0(2𝑖) (1 − 𝑞)𝑖 𝑞2−𝑖] 𝑞18

= 480 (1 − 𝑞)3 𝑞17 + 3600 (1 − 𝑞)4 𝑞16+ 13992 (1 − 𝑞)5 𝑞15 + 31200 (1 − 𝑞)6 𝑞14+ 76800 (1 − 𝑞)7 𝑞13+ 125700 (1 − 𝑞)8 𝑞12+ 167900 (1 − 𝑞)9 𝑞11+ 184750 (1 − 𝑞)10 𝑞10+ 20∑𝑖=11

(20𝑖 ) (1 − 𝑞)𝑖 𝑞20−𝑖.

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From these equations, we find that 𝐾36 does not con-tain the connected subhypergraph with 2 edges. The num-ber of spanning hypertrees is 480, the number of con-nected spanning subhypergraphs with 4 edges is 3600, andso on.

3.2. The Number of Spanning Hypertrees in 𝑟-Uniform Com-plete Hypergraphs. According to the definition of the hyper-tree in this paper and the theory of network reliability,the spanning hypertree of 𝑟-uniform complete hypergraphmeans a connected spanning subhypergraph with the min-imum number of edges. The hypergraph 𝐻1 = (𝑉, 𝜀1) is oneexample, where 𝑉 = {1, 2, 3, 4, 5}, 𝜀1 = {𝐸1 = {1, 2, 5}, 𝐸2 ={3, 4, 5}}. And the hypergraph 𝐻2 = (𝑉, 𝜀2) is another

example, where 𝑉 = {1, 2, 3, 4, 5}, 𝜀2 = {𝐸1 = {1, 4, 5}, 𝐸2 ={2, 4, 5}, {3, 4, 5}}. Both 𝐻1 and 𝐻2 are spanning subhyper-graphs of 𝐾35 , and they are all hypertrees. By Theorem 1, a3-uniform hypergraphs with 5 vertices have at least 2 edges.|𝜀1| = 2, so 𝐻1 is a spanning hypertree of 𝐾35 ; |𝜀2| = 3,so 𝐻2 is not a spanning hypertree of 𝐾35 ; it is a hypertreeand a spanning subhypergraph of 𝐾35 . Clearly, 𝜏(𝐾35) = 15and the number of spanning subhypergraphs with hypertreestructure in𝐾35 is 25.Wefind that the spanning hypertree of 𝑟-uniform complete hypergraph is a hypertree and a spanningsubhypergraph, but the reverse is not true.

Combined with the above recurrence relation of the 𝑟-uniform complete hypergraph 𝐾𝑟𝑛 (𝑛 ≥ 2, 2 ≤ 𝑟 ≤ 𝑛 − 1) andits equivalent standard conversion, we can get the number ofspanning trees in𝐾𝑟𝑛.Lemma 6. The number of spanning hypertrees in 𝐾𝑟𝑛 is givenby the following: 𝜏 (𝐾𝑟𝑛) = s⌈(𝑛−1)/(𝑟−1)⌉ (𝐾𝑟𝑛) . (6)

Proof. Let𝐻 be a connected spanning subhypergraph of 𝐾𝑟𝑛.To make 𝐻 become a spanning hypertree of 𝐾𝑟𝑛, if the edgenumber of 𝐻 denotes 𝑚(𝑆𝑇), for the Theorem 1, 𝑚(𝑆𝑇) ≥(𝑛 − 1)/(𝑟 − 1)must be first condition.

Case 1. If (𝑛−1)/(𝑟−1) is an integer, then𝑚(𝑆𝑇) = (𝑛−1)/(𝑟−1).Case 2. If 𝑛−1 = 𝑘(𝑟−1)+ 𝑡, where 𝑘 is a nonnegative integerand 𝑡 = 1, 2, ⋅ ⋅ ⋅ , 𝑟 − 2,then𝑚(𝑆𝑇) = 𝑘 + 1.

Based on Cases 1 and 2, the conclusions we need to provehave been established.

Table 2 shows the number of spanning trees of 𝑟-uniform complete hypergraphs 𝐾𝑟𝑛 of orders 4, 5, 6, 7, and8.

According to the definition of the spanning hypertreeabove and standard expression of reliability in 𝑟-uniformcomplete hypergraphs, the result of Lemma 6 is intuitive andtrue for all hypergraphs. However, it is difficult to solve allthe counting problems of the spanning hypertrees in generalhypergraphs, it needs to be furtherly explored in the futureresearch.

About enumeration of spanning hypertrees in 𝐾𝑟𝑛, forsome special cases, we get some results as follows.

Theorem7. If 1 ≤ 𝑘 ≤ ⌊𝑛/2⌋ and𝐾𝑟𝑛 (𝑟 = 𝑛−𝑘) is a 𝑟-uniformcomplete hypergraph with order 𝑛, then

(a) 𝜏 (𝐾𝑟𝑛) = ((𝑛𝑘)2 ) −((𝑛 − 1𝑘 − 1)2 )− 𝑘−1∑𝑟=1

(𝑛 − 1𝑟 ) 𝜏 (𝐾𝑛−𝑘𝑛−𝑟 ) ;(b) 𝜏 (𝐾𝑟𝑛) = 12 ( 𝑛2𝑘)(2𝑘𝑘 ) .

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Mathematical Problems in Engineering 5

Table 2: The number of spanning trees for small hypergraphs.

r n4 5 6 7 8

3 6 15 480 735 1178104 10 45 70 145605 15 105 2806 21 2107 28

Proof. Let 𝐾𝑟𝑛 (𝑟 = 𝑛 − 𝑘, 1 ≤ 𝑘 ≤ ⌊𝑛/2⌋) be a 𝑟-uni-form complete hypergraph with order 𝑛. By Lemma 6, wehave 𝜏(𝐾𝑛−𝑘𝑛 ) = s2(𝐾𝑛−𝑘𝑛 )(1 ≤ 𝑘 ≤ ⌊𝑛/2⌋). And by thestandard expression of the recursive formula of 𝑅(𝐾𝑛−𝑘𝑛 ) inTheorem 3, we can deduce s2(𝐾𝑛−𝑘𝑛 ) = ( ( 𝑛𝑘 )2 ) − ( ( 𝑛−1𝑘−1 )2 ) −∑𝑘−1𝑟=1 ( 𝑛−1𝑟 ) 𝜏(𝐾𝑛−𝑘𝑛−𝑟 ), so the first equality in the equation holds.

On the other hand, the edge number of spanning hyper-trees in 𝐾𝑛−𝑘𝑛 s is exactly equal to 2, from the perspective ofcombinatorics; the number of the spanning hypertrees 𝜏(𝐾𝑟𝑛)in𝐾𝑛−𝑘𝑛 s is (1/2) ( 𝑛2𝑘 ) ( 2𝑘𝑘 ).

So far, we have completed the proof of Theorem 7.

Remark 8. By Theorem 7 and let 𝜏(𝐾𝑛−𝑘𝑛−𝑟 ) =(1/2) ( 𝑛−𝑟2𝑘−2𝑟 ) ( 2𝑘−2𝑟𝑘−𝑟 ), we have the equation as follows:

((𝑛𝑘)2 ) −((𝑛 − 1𝑘 − 1)2 )− 12 𝑘−1∑𝑟=1(𝑛 − 1𝑟 )( 𝑛 − 𝑟2𝑘 − 2𝑟)(2𝑘 − 2𝑟𝑘 − 𝑟 )

= 12 ( 𝑛2𝑘)(2𝑘𝑘 ) .(8)

As a corollary toTheorem 7, we obtain the result that thenumber of spanning hypertrees in special cases of 𝑟-uniformcomplete hypergraphs.

Corollary 9. Let 𝐾𝑟𝑛 be a 𝑟-uniform complete hypergraphs oforder 𝑛, when 𝑟 = 𝑛 − 1, 𝑛 − 2, 𝑛 − 3

𝜏 (𝐾𝑛−1𝑛 ) = (𝑛2) (𝑛 ≥ 3) ;𝜏 (𝐾𝑛−2𝑛 ) = 3(𝑛4) (𝑛 ≥ 5) ;𝜏 (𝐾𝑛−3𝑛 ) = 10(𝑛6) (𝑛 ≥ 7) .

(9)

Proof. The transformation of the recurrence in Theorem 3is the homogeneous polynomial ∑( 𝑛𝑟 )𝑖=0 𝑠𝑖(𝐾𝑟𝑛)(1 − 𝑞)𝑖𝑞( 𝑛𝑟 )−𝑖.According to the definition of the spanning hypertree of a 𝑟-uniform complete hypergraph, we achieve some interestingconclusions as follows:

𝜏 (𝐾𝑛−1𝑛 ) = 𝑠2 (𝐾𝑛−1𝑛 ) = (𝑛2) (𝑛 ≥ 3) ;𝜏 (𝐾𝑛−2𝑛 ) = 𝑠2 (𝐾𝑛−2𝑛 )

= ((𝑛2)2 ) − (𝑛 − 12 ) − (𝑛 − 1) (𝑛 − 12 )= 3(𝑛4) (𝑛 ≥ 5) ;

𝜏 (𝐾𝑛−3𝑛 ) = 𝑠2 (𝐾𝑛−3𝑛 )= ((𝑛3)2 ) −((𝑛 − 12 )2 )− (𝑛 − 12 )(𝑛 − 22 ) − (𝑛 − 1) 𝜏 (𝐾𝑛−3𝑛−1)

= 10(𝑛6) (𝑛 ≥ 7) .

(10)

In a similar fashion as stated in Theorem 7, we can provethe following statement for 𝜏(𝐾𝑛2𝑛).Theorem 10. Let 𝐾𝑛2𝑛 be a 𝑛-uniform complete hypergraphwith order 2𝑛, then

𝜏 (𝐾𝑛2𝑛) = ((2𝑛𝑛 )3 ) −((2𝑛 − 1𝑛 )3 )− 𝑛−1∑𝑟=1

( 2𝑛 − 1𝑛 − 1 + 𝑟) 𝑆3 (𝐾𝑛𝑛+𝑟) .(11)

6 Mathematical Problems in Engineering

Remark 11. Because 𝑠2(𝐾𝑛2𝑛) = 0, so we can get

((2𝑛𝑛 )2 ) −((2𝑛 − 1𝑛 )2 )= (2𝑛 − 1𝑛 ) + 12 𝑛−1∑𝑟=1( 2𝑛 − 1𝑛 + 𝑟 − 1)(𝑛 + 𝑟2𝑟 )(2𝑟𝑟 ) .

(12)

4. Conclusions and Open Problems

In this paper, a recursive algorithm is proposed for calculatingthe all-terminal reliability of a 𝑟-uniform complete hyper-graph. Because the 2-uniform hypergraph is a graph, andthe computation of the all-terminal reliability of network isan NP-hard problem [31], we know that the computation ofreliability of hypernetwork is also NP-hard problem indi-rectly. The symmetry of the 𝐾𝑛 leads to the fact that all-terminal reliability 𝑅(𝐾𝑛, 𝑞) can be calculated in quadratictime using Theorem 2. We also use the symmetry of the 𝐾𝑟𝑛,so 𝑅(𝐾𝑟𝑛, 𝑞) can be calculated in time𝑀(𝑛2) usingTheorem 3.The algorithm presented here is a major improvement overcomplete state enumeration of common approaches.

There still exist some interesting openquestions for futureresearch in this field:

(i) How can we get a recurrence relation for the 𝐾-Terminal reliability polynomial of a 𝑟-uniform com-plete hypergraph?

(ii) How can we calculate 2-edge-connected (or higher)reliability of 𝑟-uniform complete hypergraph?

(iii) How can we calculate more about the result of thenumber of spanning hypertrees in 𝑟-uniform com-plete hypergraph?

For a 𝑟-uniform complete hypergraph 𝐾𝑟𝑛, when 𝑟 isa constant, it is very challenging to get the expression of𝜏(𝐾𝑟𝑛) which is just about 𝑛 and 𝑟 [28]. According to therecursive relations of reliability of 𝐾𝑟𝑛 proposed in this paperand combinatorics, new progress is expected to be made insolving such problems.

(iv) How can we attack the problem for general hyper-graphs?

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by NSFC (Grants nos. 11661069,61663041, and 61763041), the Science Found of QinghaiProvince (Grants nos. 2015-ZJ-723 and 2018-ZJ-718), and the

Fundamental Research Funds for the Central Universities(Grant no. 2017TS045).

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