
The aim of study is to investigate the Hirota equation which has a significant role in applied sciences, like maritime, coastal engineering, ocean, and the main source of the environmental action due to energy transportation on floating anatomical structures. The classical Hirota model has transformed into a fractional Hirota governing equation by using the space-time fractional Riemann-Liouville, time fractional Atangana-Baleanu and space-time fractional β differential operators. The most generalized new extended direct algebraic technique is applied to obtain the solitonic patterns. The utilized scheme provided a generalized class of analytical solutions, which is presented by the trigonometric, rational, exponential and hyperbolic functions. The analytical solutions which cover almost all types of soliton are obtained with Riemann-Liouville, Atangana-Baleanu and β fractional operator. The influence of the fractional-order parameter on the acquired solitary wave solutions is graphically studied. The two and three-dimensional graphical comparison between Riemann-Liouville, Atangana-Baleanu and β-fractional derivatives for the solutions of the Hirota equation is displayed by considering suitable involved parametric values with the aid of Mathematica.
Citation: Muhammad Imran Asjad, Waqas Ali Faridi, Adil Jhangeer, Maryam Aleem, Abdullahi Yusuf, Ali S. Alshomrani, Dumitru Baleanu. Nonlinear wave train in an inhomogeneous medium with the fractional theory in a plane self-focusing[J]. AIMS Mathematics, 2022, 7(5): 8290-8313. doi: 10.3934/math.2022462
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The aim of study is to investigate the Hirota equation which has a significant role in applied sciences, like maritime, coastal engineering, ocean, and the main source of the environmental action due to energy transportation on floating anatomical structures. The classical Hirota model has transformed into a fractional Hirota governing equation by using the space-time fractional Riemann-Liouville, time fractional Atangana-Baleanu and space-time fractional β differential operators. The most generalized new extended direct algebraic technique is applied to obtain the solitonic patterns. The utilized scheme provided a generalized class of analytical solutions, which is presented by the trigonometric, rational, exponential and hyperbolic functions. The analytical solutions which cover almost all types of soliton are obtained with Riemann-Liouville, Atangana-Baleanu and β fractional operator. The influence of the fractional-order parameter on the acquired solitary wave solutions is graphically studied. The two and three-dimensional graphical comparison between Riemann-Liouville, Atangana-Baleanu and β-fractional derivatives for the solutions of the Hirota equation is displayed by considering suitable involved parametric values with the aid of Mathematica.
Network reliability is an important topic in network research, network performance analysis, and combinatorial mathematics. And researchers usually use graph theoretic models to study it extensively. The network reliability can be separated into three types of models: edges are perfectly reliable while vertices survive independently with a fixed probability [9,12]; vertices are perfectly reliable while edges survive independently with a fixed probability [3,11,16]; and vertices and edges survive independently of each other with some fixed probabilities [6,8]. There are two aspects on the network reliability: reliability analysis and reliability design. The purpose of reliability analysis is to compute the reliability or unreliability polynomial of a given graph [7,19,23,24]. The purpose of reliability design is to find the graphs with the maximum reliability polynomial or the minimum unreliability polynomial among graphs with the same number of vertices and edges [1,2,3,8,10,11,14,16]. In addition, according to the number of vertices which are connected in the graph, these models can be divided into two main research categories: k-terminal reliability (the probability that k specified target vertices in a given graph are connected) [2,7,13,15,17,23,24]; and all-terminal reliability (the probability that the entire graph is connected) [1,3,8,9,10,11,14,16,19,21,22].
In practice, the network is required to run normally even if some edges are fail. If each edge of the network survives independently with a fixed probability, the network with the largest connected probability of target vertices is defined as the most reliable graph. There are more results about the most reliable graphs for all-terminal graphs, seeing [1,3,10,14,21]. However, there are a few studies on reliability analysis of k-terminal networks, which calculate the reliability polynomials of graphs [7,23,24]. And there are even fewer results on the reliability design of k-terminal networks. In 2018, Betrand et al. [2] gave some important results about the most reliable two-terminal graph, and determined several locally most reliable two-terminal graphs when the vertices are perfectly reliable and the edges survive independently with probability 0≤p≤1. And they also proved that there is no uniformly most reliable two-terminal simple graph in some graph families. It is natural to consider the following problems.
Problem. Do the three-terminal graphs have locally most reliable graph or uniformly most reliable graph as two-terminal graphs? How does one construct locally most reliable three-terminal graphs with given number of vertices and edges? Is the locally most reliable three-terminal graph also uniformly most reliable?
With these questions, this research extends the study from the two-terminal graphs to three-terminal graphs, studies the locally most reliable three-terminal simple sparse graphs (graphs with edges less than or equal to a constant multiple of the number of vertices) and considers whether the locally most reliable graph is also the uniformly most reliable graph. The structure of this paper is organized as follows. Fundamental definitions and notations are given in Section 2. In Section 3, some locally most reliable graphs are determined for three-terminal graphs with n vertices and m edges, where n≥5 and 9<m≤4n−10. Some locally most reliable graphs are further evaluated that they are not uniformly most reliable graphs, when 11<m≤3n−5 and m≡2(mod3) or 3n−5<m≤4n−10. Section 4 summarizes the results of this research.
Some basic notation is list here. For integers a,b and r, the notation a≡r(modb) indicates that the reminder of a divided by b is r, and ⌊ab⌋ is the largest integer not greater than ab. In this paper, we will only consider simple graphs in which there are no multiple edges and loops. In a graph G, the degree of the vertex v is the number of edges incident with v, denoted by d(v). The complete graph on n vertices is denoted by Kn, and K1,n denotes the simple graph on n+1 vertices with one vertex of degree n and n vertices of degree 1. The union of graphs G and H is the graph with vertex set V(G)∪V(H) and edge set E(G)∪E(H), which is denoted by G∪H. If l is a positive integer, then l⋅G denotes the disjoint union of l copies G. The join of G and H, which is denoted by G∨H, has vertex set V(G)∪V(H) and edge set E(G)∪E(H)∪{uv|u∈G,v∈H}. Suppose u and v are two vertices in G, G∪{uv} is the graph obtained by adding an edge between u and v to G, and G−{uv} is the graph obtained by deleting the edge between u and v from graph G. For notation and terminology not defined in this paper we follow [4].
A three-terminal graph is an undirected and simple graph G=(V(G),E(G)) with three specified target vertices r,s and t in V(G). Using Gn,m denotes the set of all three-terminal graphs on n vertices and m edges. The probability that three specified target vertices r,s,t of a graph G∈Gn,m remain connected when each of its edges survives independently with probability p is called the three-terminal reliability (or the three-terminal reliability polynomial) of G. The rst-subgraph of G is the subgraph of G such that r,s,t are connected with each other. Let Ni(G) (simply Ni) be the number of rst-subgraphs with i edges of graph G, then the three-terminal reliability polynomial of the graph G∈Gn,m can be defined as
R3(G;p)=m∑i=2Nipi(1−p)m−i. |
For some three-terminal graphs with small number of vertices and edges, we can compute the reliability polynomial directly by definition. However, for a three-terminal graph with many vertices and edges, the number of rst-subgraphs with i edges is very large, which possibly leads to the loss or repetition of some rst-subgraphs in the process of finding rst-subgraphs. So, it is very difficult to determine the coefficients of the reliability polynomial by the definition. In order to obtain the reliability polynomial, it is necessary to use the Factorization approach as indicated in the following lemma.
Lemma 2.1. ([18]) For any edge e in a three-terminal graph G∈Gn,m, the following factorization holds:
R3(G;p)=pR3(G⋅e;p)+(1−p)R3(G−e;p), |
where G⋅e is the graph obtained by contracting the endpoints of edge e in G and G−e is the graph obtained by deleting the edge e from G.
Example 1. Figure 1 depicts two special three-terminal graphs in G7,14 with three target vertices r,s,t. Each edge of these graphs survives independently with probability p. By definition, we have
R3(G1;p)=14∑i=2Ni(G1)pi(1−p)14−i, R3(G2;p) = 14∑i=2Ni(G2)pi(1−p)14−i.
By calculation, we get
R3(G1;p)−R3(G2;p) = 52p14−490p13+2039p12−4898p11 + 7433p10−7288p9+4463p8−1464p7 + 63p6+122p5−34p4+2p3.
Figure 2 gives a plot of R3(G1;p)−R3(G2;p). Clearly, G1 is more reliable than G2 for p→0 (the value of p sufficiently close to 0) and for p→1 (the value of p sufficiently close to 1). In fact, by Theorems 3.3 and 3.4, it is easy to see that when n=7 and m=14, G1≅B7,7 and G1 is the locally most reliable graph in G7,14 for p→0 and for p→1. However, when p is in the range (0.1,0.65), G2 is more reliable than G1. Thus, there is no uniformly most reliable graph in G7,14.
According to the definitions of the locally most reliable all-terminal graph [1] and the uniformly most reliable two-terminal graph [2], we defined the locally most reliable graph and the uniformly most reliable graph for three terminal graphs.
Definition 2.1. For p0=0 (or 1), if there is an ε>0 such that R3(G;p)≥R3(H;p) for all H∈Gn,m and for all p∈[0,1]∩(p0−ε,p0+ε), then G is the locally most reliable graph in Gn,m for p→0 (or for p→1). In particular, a graph G is the uniformly most reliable graph in Gn,m, if R3(G;p)≥R3(H;p) for all H∈Gn,m and all 0≤p≤1.
In this section, we determine some locally most reliable graphs in Gn,m for 9≤m≤4n−10. Then we also consider whether these locally most reliable graphs are the uniformly most reliable graphs. An rst-subgraph with i edges is minimal if it does not contain any rst-subgraphs with edges number less than i, otherwise it is non-minimal. An rst-cutset is a set of edges, whose removal makes at least two target vertices disconnected. And the rst-edge connectivity of G is the smallest size of an rst-cutset, denoted by λ(rst) or simply λ. It is difficult to find the exact cases that three target vertices are connected, which is NP-complete [20]. Therefore, we determine the most reliable graph by the following lemma.
Lemma 3.1.([1]) Let G,H∈Gn,m, the three-terminal reliability polynomial of G and H is
R3(G;p)=m∑i=2Ni(G)pi(1−p)m−i and R3(H;p)=m∑i=2Ni(H)pi(1−p)m−i, respectively.
Suppose there exist integers k and l, such that Ni(G)=Ni(H) for 2≤i<k or for l<i≤m. Then
(1) If Nk(G)>Nk(H), then R3(G;p)>R3(H;p) for p→0,
(2) If Nl(G)>Nl(H), then R3(G;p)>R3(H;p) for p→1.
From Lemma 3.1, we see that if G∈Gn,m is the locally most reliable graph for p→0, then it must contain the triangle rst and N3 is the largest among graphs containing the triangle rst in Gn,m. It is not hard to see that Ni=(mi) for m−λ+1≤i≤m and Nm−λ=(mλ)−a, where a is the number of the rst-cutsets of size λ. Then for p→1, if G is the locally most reliable graph in Gn,m, then it must have the largest rst-edge connectivity λ, the number of rst-cutsets of size λ attains the minimum value.
We first introduce some important graphs which will be used below.
Let n≥4 and 0≤l≤n−4 be integers. The three-terminal graph with n vertices, which is drawn as Figure 3, is denoted by An,l, where the vertex set V(An,l) is {r=v1,s=v2,t=v3,v4,⋯,vn} and the edge set E(An,l) contains the following 3n−6+l edges:
{rs,rt,st,vivjwherei∈{1,2,3},4≤j≤n,v4vjwhere5≤j≤l+4. |
Let n≥4 and 4≤w≤n be integers. The three-terminal graph with n vertices, which is drawn as Figure 4, is denoted by Bn,w, where the vertex set V(Bn,w) is {r=v1, s=v2, t=v3, v4, ⋯, vw, ⋯, vn} and the edge set E(Bn,w) contains the following 3w−7 edges:
{rs,rt,st,vivjwherei∈{1,2,3},4≤j≤w−1,vivwwherei∈{1,2}. |
Theorem 3.1. Let n≥5 and 9≤m≤3n−5 be integers and m≡0 or 1(mod3). Then the graph
A⌊m3⌋+2,m−3⌊m3⌋∪(n−⌊m3⌋−2)⋅K1 |
is the locally most reliable graph in Gn,m for p→0.
Proof. Assume that n and m satisfy the given conditions and let G be the locally most reliable graph in Gn,m for p→0. By Lemma 3.1, we see that G must contain the triangle rst and N3 is the largest among graphs containing the triangle rst in Gn,m.
According to the rst-subgraph containing the edge number of triangle rst, the rst-subgraph with 3 edges are consisted of the following four cases:
Case 1. All of three edges are from the triangle rst, saying rs,rt,st.
Case 2. Two of three edges are from the triangle rst and another edge not in the triangle, saying rs,st,vivj, where 1≤i≤n,4≤j≠i≤n.
Case 3. One of three edges is from the triangle rst and other two edges are not in the triangle, saying rvi,vis,rt, where 4≤i≤n.
Case 4. None of three edges is from the triangle rst, then they are vir, vis, vit, where 4≤i≤n.
Note that the number of the rst-subgraphs in Cases 1 and 2 is 1 and 3(m−3), respectively. N3 attains the maximum value if and only if the number of the rst-subgraphs of both Case 3 and Case 4 attains maximum value. By calculation, if the number of the rst-subgraphs in Case 4 attains maximum value, then E(G) must contain ⌊m−33⌋=⌊m3⌋−1 edge sets {vir,vis,vit} (4≤i≤⌊m3⌋+2). Since m≡0 or 1(mod3), the number of the rst-subgraphs in Case 3 attains the maximum value, while it is maximum in Case 4. Therefore we get the following.
If m≡0(mod3), then E(G)={rs,rt,st}∪{vivj|1≤i≤3,4≤j≤m3+2}, and G is Am3+2,0 ∪ (n−m3−2)⋅K1.
If m≡1(mod3), the edge set of G is consisted of the triangle rst, m−43 edge subsets as {vir,vis,vit} (4≤i≤m+53), and the remaining edge either joining one target vertex and one non-target vertex or connecting two non-target vertices. For convenience, the remaining edge is denoted by e. If 9≤m≤3n−11, then the degrees of non-target vertices in G−e are 3 or 0. There are four possible joining types for e. The first type is to connect two non-target vertices of degree 3, without losing generality setting e=v4v5, and the final graph is denoted by G1. The second type is to join one non-target vertex of degree 3 and the other non-target vertex of degree 0, without losing generality setting e=v4vm+83, and the final graph is denoted by G2. The third type is to connect two non-target vertices of degree 0, without losing generality setting e=vm+83vm+113, and the final graph is denoted by G3. The fourth type is to join one target vertex and one non-target vertex of degree 0, without losing generality setting e=rv(m+8)/3, and the final graph is denoted by G4. By calculation, N4(G1)>N4(Gi) for i∈{2,3,4}. By Lemma 3.1, E(G)=E(G1)={rs,rt,st,v4v5}∪{vivj|1≤i≤3,4≤j≤m+53}, which implies that G≅Am+53,1∪(n−m+53)⋅K1. If 3n−11<m≤3n−8, then there are n−4 non-target vertices of degree 3 and only one non-target vertex of degree 0 in G−e. There are three possible types for e and the final graph is in {G1,G2,G4}. By calculation and comparison, we can see that G≅Am+53,1∪(n−m+53)⋅K1. If 3n−8<m≤3n−5, then the degrees of all non-target vertices in G−e are equal to 3. Then e is only one type to connect two non-target vertices of degree 3, which means that G≅G1. So, G≅Am+53,1∪(n−m+53)⋅K1.
From the above argument, we see that A⌊m3⌋+2,m−3⌊m3⌋∪(n−⌊m3⌋−2)⋅K1 is the locally most reliable graph in Gn,m for p→0.
Theorem 3.2. Let n≥5 and 9≤m≤3n−5 be integers and m≡0 or 1(mod3). Then the graph
A⌊m3⌋+2,m−3⌊m3⌋∪(n−⌊m3⌋−2)⋅K1 |
is the locally most reliable graph in Gn,m for p→1.
Proof. Let A=A⌊m3⌋+2,m−3⌊m3⌋∪(n−⌊m3⌋−2)⋅K1. Let G∈Gn,m be the locally most reliable graph for p→1. Then by Lemma 3.1, G must have the largest rst-edge connectivity λ, and the number of rst-cutsets of size λ attains the minimum value among graphs with the largest λ.
Obviously, λ≤min{d(r),d(s),d(t)}≤⌊m3⌋+1. If {rs,rt,st}∪{vivj|1≤i≤3,4≤j≤⌊m3⌋+2}⊆E(G), then min{d(r),d(s),d(t)}=⌊m3⌋+1. Let C be the minimal rst-cutset of G, then there must exist a component containing just one target vertex and k1 (0≤k1≤n−3) non-target vertices ui(1≤i≤k1) in G−C, where u1,u2,⋯,uk∈{vi∣4≤i≤⌊m3⌋+2}, and without loss of generality, setting this target vertex as r. Then the number of edges in C is at least ⌊m3⌋+1−k+2k=⌊m3⌋+1+k≥⌊m3⌋+1. Thus λ≥⌊m3⌋+1. Hence, λ can arrive at the maximum value ⌊m3⌋+1 if {rs,rt,st}∪{vivj|1≤i≤3,4≤j≤⌊m3⌋+2}⊆E(G). Then λ=⌊m3⌋+1.
If m≡0(mod3), then λ is m3+1, d(r)=d(s)=d(t)=m3+1, r,s and t are adjacent with each other. If there is a non-target vertex v∈V(G) with d(v)≠0 or 3, then we have either λ(G)<m3+1, or Nm−(m3+1)(G)<(mλ)−3. For each non-target vertex v∈V(G), if d(v)=0 or 3, then Nm−λ(G)=(mλ)−3. Since G is the locally most reliable graph, by Lemma 3.1, Nm−λ must be maximum, then the degree of each non-target vertex is either 0 or 3. Thus, G is A.
If m≡1(mod3), then λ is m+23, and {rs,rt,st}∩E(G)≥2. When {rs,rt,st}∩E(G)=3, similarly, we can find that there are four graphs with λ=m+23 and Nm−λ=(mλ)−3, which are A, A∪{rvn}−{v4v5}, A∪{v4vn}−{v4v5}, A∪{vn−1vn}−{v4v5}, where the second and third graphs only occur when m≤3n−8 and the last only occurs when m≤3n−11. By calculation, the values of Nm−λ−1 of these four graphs is (mλ+1)−3m+12, (mλ+1)−3m+6, (mλ+1)−3m+6 and (mλ+1)−3m+6, respectively. Obviously, (mλ+1)−3m+12>(mλ+1)−3m+6, by Lemma 3.1, G is A. When {rs,rt,st}∩E(G)=2, similarly, by calculation, we find that for all graphs with λ=m+23, there is Nm−λ<(mλ)−3. Therefore, by Lemma 3.1, if m≡1(mod3), then {rs,rt,st}∩E(G)=3 and G is A.
Therefore, the graph A⌊m3⌋+2,m−3⌊m3⌋∪(n−⌊m3⌋−2)⋅K1 is the locally most reliable graph in Gn,m for p→1.
Theorems 3.1 and 3.2 show that when n≥5, 9≤m≤3n−5 and m≡0 or 1(mod3), A⌊m3⌋+2,m−3⌊m3⌋ ∪(n−⌊m3⌋−2)⋅K1 is the locally most reliable graph in Gn,m for both p→0 and p→1. If m≡2(mod3), we have the following theorems, whose proofs are similar to the proofs of Theorems 3.1 and 3.2.
Theorem 3.3. Let n≥5 and 9≤m≤3n−5 be integers and m≡2(mod3). Then the graph
Bn,⌊m3⌋+3∪(n−⌊m3⌋−3)⋅K1 |
is the locally most reliable graph in Gn,m for p→0.
Theorem 3.4. Let n≥5 and 9≤m≤3n−5 be integers and m≡2(mod3). Then the graph
Bn,⌊m3⌋+3∪(n−⌊m3⌋−3)⋅K1 |
is the locally most reliable graph in Gn,m for p→1.
Theorems 3.3 and 3.4 show that when n≥5, 9≤m≤3n−5 and m≡2(mod3), Bn,⌊m3⌋+3 ∪(n−⌊m3⌋−3)⋅K1 is the locally most reliable graph in Gn,m for both p→0 and p→1. Is it the uniformly most reliable graph for 11<m≤3n−5 (n≥5)? In order to solve this problem, we need to compute the reliability polynomials of some three-terminal graphs.
Lemma 3.2. Let n≥4 be an integer. Then
R3(An,0;p)=1−(4p6−18p5+30p4−20p3+6p−2)(1−3p2+2p3)n−4−3(1−p)2(1−2p2+p3)n−3. |
Proof. The vertices in An,0 are labeled same as Figure 3. By Lemma 2.1, we can calculate a recurrence relation for the three-terminal probability polynomial of An,0.
R3(An,0;p)=p3R3(G1;p)+p3(1−p)R3(G2;p) + p2(1−p)2R3(G3;p) + p3(1−p)R3(G4;p)+p2(1−p)2R3(G5;p) + p(1−p)2R3(G6;p)+p3(1−p)R3(G7;p)+p2(1−p)2R3(G8;p) + p(1−p)2R3(G9;p)+p(1−p)2R3(G10;p) + (1−p)3R3(G11;p), where the forms and reliability polynomials of Gi (1≤i≤11) are shown in Table 1.
Graph Gi | Reliability polynomial of Gi |
G1=An,0⋅v1vn⋅v1v2⋅v1v3 | 1 |
G2=An,0⋅v1vn⋅v1v2−v1v3⋅v2v3 | 1 |
G3=An,0⋅v1vn⋅v1v2−v1v3−v2v3 | 1−(1−p)(1−2p2+p3)n−4 |
G4=An,0⋅v1vn−v1v2⋅v1v3⋅v2v3 | 1 |
G5=An,0⋅v1vn−v1v2⋅v1v3−v2v3 | 1−(1−p)(1−2p2+p3)n−4 |
G6=An,0⋅v1vn−v1v2−v1v3 | R3(An−1,0;p) |
G7=An,0−v1vn⋅v2vn⋅v2v3⋅v1v2 | 1 |
G8=An,0−v1vn⋅v2vn⋅v2v3−v1v2 | 1−(1−p)(1−2p2+p3)n−4 |
G9=An,0−v1vn⋅v2vn−v2v3 | R3(An−1,0;p) |
G10=An,0−v1vn−v2vn⋅v3vn | R3(An−1,0;p) |
G11=An,0−v1vn−v2vn−v3vn | R3(An−1,0;p) |
By Table 1, we have R3(An,0;p)=(1+2p)(1−p)2R3(An−1,0;p)+p3(4−3p)+3p2(1−p)2[1−(1−p)(1−2p2+p3)n−4].
Calculating the linear non-homogeneous recurrence relation, we have
R3(An,0;p)=(1+2p)n−4(1−p)2n−8R3(A4,0;p)+1−(1+2p)n−4(1−p)2n−81−(1+2p)(1−p)2(3p2−2p3)−3p2(1−p)31−((1+2p)(1−p)2/1−2p2+p3)n−41−[(1+2p)(1−p)2/1−2p2+p3]=(−4p6+15p5−18p4+5p3+3p2)(1−3p2+2p3)n−4+[1−(1−3p2+2p3)n−4]−3(1−p)2(1−2p2+p3)n−3[1−(1−(3p2+2p3)/(1−2p2+p3))n−4]=1−(4p6−18p5+30p4−20p3+6p−2)(1−3p2+2p3)n−4−3(1−p)2(1−2p2+p3)n−3. |
The proof is completed.
Similarly as Lemma 3.2, we can get Lemmas 3.3 and 3.4.
Lemma 3.3. Let n≥6 and 4≤w≤n be integers. Then
R3(Bn,w;p)=p3+p2(1−p)[1−(1−p)(1−2p2+p3)w−4]+(p+1)(1−p)Aw−1,0. |
Lemma 3.4. Let n≥6 be an integer. Then
R3(An,2;p)=3p10−24p9+80p8−138p7+120p6−30p5−28p4+18p3 - (18p12−159p11+603p10−1272p9+1602p8−1173p7+399p6 + 42p5−78p4+18p3)(1−2p2+p3)n−6−(3p10−24p9+81p8−150p7 + 165p6−108p5+39p4−6p3)An−3,0+(p8−12p7 + 45p6−80p5+75p4−36p3+7p2)An−2,0 + (2p5−8p4+12p3−8p2+2p)An−1,0+(p2−2p+1)An,0.
With the above lemmas, we can get Theorem 3.5.
Theorem 3.5. Let n≥5 and 11<m≤3n−5 be integers and m≡2(mod3). Then the graph A⌊m3⌋+2,2∪(n−⌊m3⌋−2)⋅K1 is more reliable than Bn,⌊m3⌋+3∪(n−⌊m3⌋−3)⋅K1 in Gn,m for p=1/2.
Proof. For the convenience, let w=⌊m3⌋+3. By Lemma 3.2, we have
R3(An,0;1/2)=1+(1/2)n−1−(3/4)⋅(5/8)n−3.
By Lemmas 3.3 and 3.4 and R3(An,0;1/2), we have
R3(Bn,w;1/2)=14−116⋅(5/8)w−4+34R3(Aw−1,0;1/2)=1−(5/8)w−3+34⋅(1/2)w−2, |
R3(Aw−1,2;1/2)=6431024−364⋅(5/8)w−7+91024R3(Aw−4,0;1/2)+3256R3(Aw−3,0;1/2)+116R3(Aw−2,0;1/2)+14R3(Aw−1,0;1/2)=1−5794096⋅(5/8)w−7+831024⋅(1/2)w−5. |
Thus, R3(Aw−1,2;1/2)−R3(Bn,w;1/2)=14096[46⋅(5/8)w−7−13⋅(1/2)w−7].
Since 46⋅(5/8)w−7>46⋅(1/2)w−7>13⋅(1/2)w−7 (w=⌊m3⌋+3≥7),
R3(Aw−1,2;1/2)−R3(Bn,w;1/2)>0, which means, R3(Aw−1,2;1/2)>R3(Bn,w;1/2).
The proof is completed.
As a straightforward consequence of Theorems 3.3 or 3.4 and 3.5, we obtain the following result.
Theorem 3.6. Let n and m be integers. If n≥5, 11<m≤3n−5 and m≡2(mod3), then there is no uniformly most reliable graph in Gn,m.
Now, the existence of uniformly most reliable graph with edges less than 3n−5 is solved partly. How about the same question for a little more edges?
Lemma 3.5.([5]) Let n≥1 and 0≤m≤n−1 be integers.
If m≠3, then the unique simple graph on n vertices and m edges with the maximum number of paths of length 2 is K1,m∪(n−m−1)⋅K1.
If m=3, there are two simple graphs with the maximum number of paths of length 2: K3∪(n−3)⋅K1 and K1,3∪(n−4)⋅K1.
Theorem 3.7. Let n≥7 and 3n−5<m≤4n−10 be integers. Then the graph An,m−3n+6 is the locally most reliable graph in Gn,m for p→0.
Proof. Let G∈Gn,m be the locally most reliable graph for p→0. By Lemma 3.1 and the proof of Theorem 3.1, it is easy to see that G must contain the triangle rst and n−3 edge sets {rvi,svi,tvi} (4≤i≤n). Thus, we need to determine the remaining l=m−3n+6 edges between non-target vertices. For convenience, using ˆG denotes the subgraph of G induced by all the non-target vertices, then E(ˆG)=l. Since the different structures of ˆG may lead different Ni when i≥4, we begin with N4, which is the number of rst-subgraphs with 4 edges.
The rst-subgraphs with 4 edges of G can be divided into two cases. Some of them is minimal and others is non-minimal. There are three forms of the minimal rst-subgraph with 4 edges, which are {svi,vit,svj,vjr}, {svi,vivj,vjt,sr}, and {svi,vivj,vjt,vjr} (4≤i,j≤n,i≠j). The number of these three edge sets is 6(n−32), 12l and 6l, respectively. We can see that the number of the minimal rst-subgraphs with 4 edges is affected by l, regardless of the structure of ˆG.
The non-minimal rst-subgraph with 4 edges of G include the following cases:
C1: the minimal rst-subgraph with 2 edges,
C2: the minimal rst-subgraph with 3 edges but no minimal rst-subgraph with 2 edges.
By calculation, the number of the non-minimal rst-subgraphs with 4 edges in C1 and C2 is 3(m−32)+(m−3) and (n−3)(m−3)+6(n−3)(m−6), respectively. Then the number of the non-minimal rst-subgraphs with 4 edges is a constant for given n and m.
Therefore, whatever the structure of ˆG is, N4 is a constant for given n and m. Then we need to consider N5, which is the number of rst-subgraphs with 5 edges.
The rst-subgraphs with 5 edges of G can be divided into two cases. Some of them is minimal and others is non-minimal. The non-minimal rst-subgraph with 5 edges of G include the following cases:
D1: the minimal rst-subgraph with 2 edges,
D2: the minimal rst-subgraph with 3 edges but no minimal rst-subgraph with 2 edges,
D3: the minimal rst-subgraph with 4 edges but no minimal rst-subgraph with less than 4 edges.
By calculation, the number of the non-minimal rst-subgraphs with 5 edges in D1, D2 and D3 is 3(m−33)+(m−32), 7(n−3)(m−62)+3(n−3)(m−6)−12(n−32) and 18l(m−10)+6l+6(m−9)(n−32), respectively. Then the number of the non-minimal rst-subgraphs with 5 edges is a constant for given n and m.
There are four forms of the minimal rst-subgraph with 5 edges, which are {svi,vivj,vjr,rvk, vkt}, {svi,vivj,vjvk,vkt,rt}, {svi,vivj,vjr,vjvk,vkt}, and {rvi,svi,vivj,vjvk,vkt} (4≤i, j,k≤n,i≠j≠k). By calculation, the number of the first edge set is 12l(n−5), which is affected by l, regardless of the structure of ˆG. But the number of other three cases affected by the number of P3 in ˆG. By Lemma 3.5, if l≠3 and l≤n−4, then the number of P3 in ˆG is maximum if ˆG is K1,l∪(n−l−4)⋅K1, and G is K3∨(K1,l∪(n−l−4)⋅K1). If l=3, the number of P3 in ˆG is maximum only if ˆG is either K3∪(n−6)⋅K1 or K1,3∪(n−7)⋅K1, and G is either K3∨(K3∪(n−6)⋅K1) or K3∨(K1,3∪(n−7)⋅K1). Then by Lemma 3.1, we need to compare N6(K3∨(K3∪(n−6)⋅K1)) and N6(K3∨(K1,3∪(n−7)⋅K1)). According to the calculation method of N4 and N5, we can get that the difference of coefficient N6s of the front two graphs is −72. Then for p→0, G is K3∨(K1,3∪(n−7)⋅K1).
From the above argument, we conclude that the graph An,m−3n+6 is the locally most reliable graph in Gn,m for p→0.
Now, two classes of graphs are given, which will be used in the following theorems.
Let n≥4 and 0≤l≤⌊n−32⌋ be integers. The three-terminal graph with n vertices, which is drawn as Figure 5, is denoted by Cn,l, where the vertex set V(Cn,l) is {r=v1,s=v2,t=v3,v4,⋯,vn} and the edge set E(Cn,l) contains the following 3n−9+l edges:
{rs,rt,st,vivjwherei∈{1,2,3},4≤j≤n,v2iv2i+1where2≤i≤l+1. |
Let n≥4 and 0≤l≤n−4 be integers. The three-terminal graph with n vertices, which is drawn as Figure 5, is denoted by C′n,l, where the vertex set V(C′n,l) is {r=v1,s=v2,t=v3,v4,⋯,vn} and the edge set E(C′n,l) contains the following 3n−9+l edges:
{rs,rt,st,vivjwherei∈{1,2,3},4≤j≤n,vjvj+1where4≤j≤l+3. |
Theorem 3.8. Let n≥7 and 3n−5<m≤3n−6+⌊n−32⌋ be integers. Then the graph Cn,m−3n+6 is the unique locally most reliable graph in Gn,m for p→1.
Proof. Let G∈Gn,m be the unique locally most reliable graph for p→1. Then by Lemma 3.1, the value of the rst-edge connectivity λ of G must be as large as possible.
let C be the minimal rst-cutset of G, then there must exist a component containing just one target vertex and k (0≤k≤n−3) non-target vertices ui (1≤i≤k) in G−C, without loss of generality, setting this target vertex as r. Clearly, λ≤min{d(r),d(s),d(t)}≤n−1. If d(r)=d(s)=d(t)=n−1, then |C|≥d(r)−k+2k=d(r)+k≥n−1. Hence, λ can arrive at the maximum value n−1 if and only if d(r)=d(s)=d(t)=n−1. Then, G contains the triangle rst and n−3 edge sets as {rvi,svi,tvi} (4≤i≤n). These 3n−6 edges are confirmed, we also need to determine the remaining m−3n+6 edges connecting non-target vertices.
By Lemma 3.1, we need to compare the number of rst-subgraph with m−n+1 edges, which is denoted as Nm−n+1, of graphs with λ=n−1. Continue to calculate the minimal rst-cutset of G, |C|=d(r)−k+k∑i=1[d(ui)−1]−2m′=n−2k−2m′−1+k∑i=1d(ui), where m′ is the number of edges between these k non-target vertices. It is clear to see that k∑i=1d(ui)≥3k+2m′, thus |C|≥n+k−1. Then, we can get that the component of k+1 vertices generated by deleting the minimal rst-cutset of size n−1 contains only the target vertex. Thus, we have Nm−n+1=(mn−1)−3, which is a constant for given n and m. By Lemma 3.1, we need to consider Nm−n.
The component of k+1 vertices generated by deleting the minimal rst-cutset of size n contains one target vertex and one non-target vertex of degree 3. Thus, we have Nm−n=(mn)−3(m−n+11)−3(a1), where a is the number of non-target vertices with degree 3. Since Cn,m−3n+6 has the fewest non-target vertices with degree 3 in graphs with λ=n−1, Nm−n(Cn,m−3n+6) gets the maximum value.
Therefore, Cn,m−3n+6 is the locally most reliable graph in Gn,m for p→1.
Theorem 3.9. Let n≥7 and 3n−6+⌊n−32⌋<m≤4n−10 be integers. Then the graph C′n,m−3n+6 is more reliable than An,m−3n+6 in Gn,m for p→1.
Proof. For convenience, let l=m−3n+6. In An,l, there are n−4−l vertices of degree 3, l vertices of degree 4, a vertex of degree l+3 and 3 target vertices of degree n−1. And C′n,l has n−4−l vertices of degree 3, 2 vertices of degree 4, l−1 vertices of degree 5 and 3 target vertices of degree n−1. The rst-edge connectivity λ of An,l and C′n,l are the same, where λ=n−1.
It is easy to calculate that
Nm−j(An,l)=Nm−j(C′n,l)=(mj) (0≤j≤n−2);
Nm−λ(An,l)=Nm−λ(C′n,l)=(mλ)−3;
Nm−λ−1(An,l)=Nm−n(An,l)=Nm−λ−1(C′n,l)=(mn)−3(m−l−3);
and
Nm−λ−2(An,l)=Nm−n−1(An,l)=(mn+1)−3(n−4−l2)−3(m−n+12)−3(n−4−l)(m−n−1)−3l; |
Nm−λ−2(C′n,l)=Nm−n−1(C′n,l)=(mn+1)−3(n−4−l2)−3(m−n+12)−3(n−4−l)(m−n−1)−6. |
Since l=m−3n+6≥3, Nm−λ−2(C′n,l)>Nm−λ−2(An,l).
By the Lemma 3.1, C′n,m−3n+6 is more reliable than An,m−3n+6 in Gn,m for p→1.
We give the locally most reliable graph in Gn,m with 3n−5<m≤3n−6+⌊n−32⌋ (n≥7) for p→1, as shown in Theorem 3.8. If 3n−6+⌊n−32⌋<m≤4n−10 (n≥7), we construct a graph with m edges that is more reliable than An,m−3n+6 for p→1, as shown in Theorem 3.9. Thus, we obtain the following result.
Theorem 3.10. Let n and m be integers. If n≥7 and 3n−5<m≤4n−10, then there is no uniformly most reliable graph in Gn,m.
This research focuses on characterizing the locally most reliable graph for three-terminal spare graphs. There is rare literature on the locally most reliable graph for three-terminal graphs. Based on the results of this research, the following conclusions can be drawn.
If 9≤m≤3n−5 (n≥5) and m≡2(mod3), the locally most reliable graph for p→0 and p→1 are determined with theoretical proofs. It is also proved that there is no uniformly most reliable three-terminal graph when 11<m≤3n−5 (n≥5) and m≡2(mod3).
The locally most reliable graph in Gn,m for p→0 is determined with proofs when 3n−5<m≤4n−10 (n≥7). The locally most reliable graph in Gn,m for p→1 for 3n−5<m≤3n−6+⌊n−32⌋ (n≥7) is also determined with proofs. Additionally, it is proved that there is no uniformly most reliable three-terminal graph when 3n−5<m≤4n−10 (n≥7).
If 9≤m≤3n−5 (n≥5) and m≡0 or 1(mod3), as shown in Theorems 3.1 and 3.2, the locally most reliable graphs for p→0 is also locally most reliable for p→1. However, it is still unknown whether the locally most reliable graph is the uniformly most reliable graph for 9≤m≤3n−5 (n≥5) and m≡0 or 1(mod3) for all 0≤p≤1.
The results of the research can be useful for designing highly reliable networks which have three target vertices. The findings of this research provide guiding significance for determining the locally most reliable graphs for general k-terminal networks.
Research supported by the National Science Foundation of China (Grant Nos. 11801296), the Tibetan Information Processing and Machine Translation Key Laboratory of Qinghai Province, the Science Found of Qinghai Province (Grant No. 2018-ZJ-718), the Key Laboratory of Tibetan Information Processing Ministry of Education, and Tibetan Information Processing Engineering Technology and Research Center of Qinghai Province.
The authors declare no conflict of interest in this paper.
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Graph Gi | Reliability polynomial of Gi |
G1=An,0⋅v1vn⋅v1v2⋅v1v3 | 1 |
G2=An,0⋅v1vn⋅v1v2−v1v3⋅v2v3 | 1 |
G3=An,0⋅v1vn⋅v1v2−v1v3−v2v3 | 1−(1−p)(1−2p2+p3)n−4 |
G4=An,0⋅v1vn−v1v2⋅v1v3⋅v2v3 | 1 |
G5=An,0⋅v1vn−v1v2⋅v1v3−v2v3 | 1−(1−p)(1−2p2+p3)n−4 |
G6=An,0⋅v1vn−v1v2−v1v3 | R3(An−1,0;p) |
G7=An,0−v1vn⋅v2vn⋅v2v3⋅v1v2 | 1 |
G8=An,0−v1vn⋅v2vn⋅v2v3−v1v2 | 1−(1−p)(1−2p2+p3)n−4 |
G9=An,0−v1vn⋅v2vn−v2v3 | R3(An−1,0;p) |
G10=An,0−v1vn−v2vn⋅v3vn | R3(An−1,0;p) |
G11=An,0−v1vn−v2vn−v3vn | R3(An−1,0;p) |
Graph Gi | Reliability polynomial of Gi |
G1=An,0⋅v1vn⋅v1v2⋅v1v3 | 1 |
G2=An,0⋅v1vn⋅v1v2−v1v3⋅v2v3 | 1 |
G3=An,0⋅v1vn⋅v1v2−v1v3−v2v3 | 1−(1−p)(1−2p2+p3)n−4 |
G4=An,0⋅v1vn−v1v2⋅v1v3⋅v2v3 | 1 |
G5=An,0⋅v1vn−v1v2⋅v1v3−v2v3 | 1−(1−p)(1−2p2+p3)n−4 |
G6=An,0⋅v1vn−v1v2−v1v3 | R3(An−1,0;p) |
G7=An,0−v1vn⋅v2vn⋅v2v3⋅v1v2 | 1 |
G8=An,0−v1vn⋅v2vn⋅v2v3−v1v2 | 1−(1−p)(1−2p2+p3)n−4 |
G9=An,0−v1vn⋅v2vn−v2v3 | R3(An−1,0;p) |
G10=An,0−v1vn−v2vn⋅v3vn | R3(An−1,0;p) |
G11=An,0−v1vn−v2vn−v3vn | R3(An−1,0;p) |