Entropy and Enumeration of Subtrees in a Cactus Network

For a given network, the number of spanning trees is a key parameter to measure its reliability in edge failure cases, while the number of subtrees is a key parameter to measure its reliability in both vertex and edge failures cases. Zhang et al. investigated the entropy of spanning trees, spanning forests and connected spanning subgraphs of Koch networks. In this paper, we extend Koch networks to 3–cactus networks, and study the entropy of subtrees of 3–cactus networks. We present a linear algorithm to count the number of subtrees in a 3–cactus network, determine the upper and lower bounds of the entropy of subtrees of these networks and characterize those attaining the extremal values. As an application, a linear algorithm is developed to count the number of subtrees in Koch networks, with complexity O(g), where g is the number of iterations. Finally, we determine the entropy of subtrees of Koch networks.

1. Introduction

The number of subtrees of a connected network is well-studied, as an important parameter to measure the reliability of a network for both vertex and edge failures. For a network G, let p be the probability of failure of a vertex in G, and q be the probability of failure of an edge in G. The reliability of the network G, denoted by R(G; p, q), is the probability that the remaining vertices can communicate with each other. Zhao [1] showed that for given numbers m and n, among the networks with n vertices and m edges, if p → 0 and q → 1 with p and 1 − q are equivalent infinitesimals, then networks with more subtrees are more reliable. Therefore, it is of interest to develop efficient algorithms to count the number of subtrees in a network and, in a given family of networks, characterize the networks with extremal number of subtrees. Denote by η(G) the number of subtrees of G. For a vertex v ∈ V(G), let η_v(G) denote the number of subtrees containing v in G. Székely and Wang [2] determined the maximum and minimum values of η(G) and characterized all extremal networks among all trees G with given number of vertices. Yan and Yeh [3] developed a linear algorithm to count the number of subtrees of a tree. In the same paper, they also characterized all trees having diameter at least d with maximum number of subtrees, as well as all trees having maximum degree at least Δ with minimum number of subtrees. On the other hand, Kirk and Wang [4] characterized all trees with a given maximum degree that have the maximum number of subtrees. Zhang and Zhang [5] and Zhang et al. [6, 7] characterized the trees attaining the maximum and minimum numbers of subtrees among all trees with the given degree sequence. Xiao et al. [8] showed that 5 + n + 2ⁿ⁻³ ≤ η(T) ≤ 2ⁿ⁻¹ + n − 1 for any tree T with n vertices. Yang et al. [9, 10] studied the number of subtrees in spiro and polyphenyl hexagonal chains and hexagonal and phenylene chains, respectively. For more results on this topic, we refer the readers to references [11–17].

Koch networks, introduced by Zhang et al. [18], are typical fractal networks. Zhang et al. showed that Koch networks have some important properties in real life networks, such as a power law degree distribution with exponent between 2 and 3, a large clustering coefficient and a small average path length. However, the number of subtrees in Koch networks has not been determined. In this paper, we investigate the entropy and enumeration of subtrees in 3–cactus networks, which is a generalization of Koch networks. We first establish a linear algorithm to count the number of subtrees in an arbitrary 3–cactus network. Then we characterize the 3–cactus networks with upper and lower bounds of the entropy of subtrees. Finally, as an application, we obtain the entropy of subtrees in Koch networks.

2. Preliminaries

We first introduce some general notation and definitions that will be used throughout the paper. Undefined notation and terminology will follow Bondy and Murty [19].

We follow Zhang et al. [18] and Wu et al. [20] to define Koch networks. Denote by G_m,g the Koch network after g iterations, which has N_g vertices, E_g edges and L_g triangles. At g = 0, G_{m, 0} is a triangle with three vertices labeled by X, Y, and Z, respectively. This triangle is called the initial triangle, and its three vertices are called hub vertices. For g ≥ 1, G_m,g is obtained from G_m,g−1 by attaching m triangles to each of the three vertices in each triangle in G_m,g−1. The Koch networks for m = 2 are shown in Figure 1.

FIGURE 1

Figure 1. The Koch networks for m = 2.

By definition, L₀ = 1 and for g ≥ 1, L_g = (3m + 1)L_g−1. This implies that $L_g={(3m+1)}^g$. Likewise, it is routine to derive that $N_g=2{(3m+1)}^g+1$ and $E_g=3{(3m+1)}^g$. More properties of Koch networks can be found in Zhang et al. [18].

To extend the notion of Koch networks, we introduce 3–cactus networks.

Definition 1. Let ${G}$(t) be the set of all 3–cactus networks with t triangles. We define a 3–cactus network G_t ∈ ${G}$(t) by the following recursive iterations.

Step 1: When t = 0, G₀ is isomorphic to K₁, the graph with one vertex and no edges. The only vertex in G₀ is called the initial vertex of G_t.

Step 2: Assume that t ≥ 1 and a 3–cactus network G_t−1 is already generated by (t−1) iterations. Then, a 3–cactus network G_t can be obtained from G_t−1 and a new triangle by identifying a vertex of the new triangle and a vertex of the G_t−1.

Let G be a subnetwork of G′. If the vertex set (or edge set) of G is a proper subset of the vertex set (or edge set) of G′, G is called a proper subnetwork of G′. Let d_G(u) denote the degree of vertex u in G, and (v₁, v₂, v₃) denote the triangle with vertices v₁, v₂, and v₃. A triangle (v₁, v₂, v₃) in G is called a d-pendant triangle if two of the vertices v₁, v₂, v₃ have degree 2 in G and the third vertex has degree equals d in G for some integer d ≥ 3. A d-pendant triangle is also called a pendant triangle.

By Definition 1, a network G_t ∈ ${G}$(t) will contain a pendant triangle with a vertex of degree at least 4. Define a 3–cactus networks G_t ∈ ${G}$(t) to be in Class I if G_t contains a 4-pendant triangle, and in Class II if G_t does not contain a 4-pendant triangle. The following observations follow from Definition 1.

Observation 1. Let t ≥ 0 be an integer, and let G_t ∈ ${G}$(t) be a network formed by Definition 1 using t iterations. Each of the following holds.

(i) The 3–cactus networks is a generalization of Koch networks, as building a Koch network G_m,g using g iterations amounts to (3m + 1)^g iterations in building a 3–cactus network.

(ii) G_t has t triangles, 2t + 1 vertices and 3t edges.

(iii) If (u₀, u₁, u₂) is a pendant triangle of G_t with d_Gt(u₁) = d_Gt(u₂) = 2, then t ≥ 2 and u₀ is a cut vertex of G_t.

(iv) Every Class II 3–cactus network G contains two pendant triangles (u₀, u₁, u₂) and (u₀, v₁, v₂) sharing a common vertex u₀.

Proof: It suffices to justify (iv) as all others are direct consequences from Definition 1. Let G be a Class II 3–cactus network. By Definition 1, G ∈ ${G}$(t) for some t ≥ 3, and if G ∈ ${G}$(3), then G has a vertex of degree 6 and all other vertices of degree 2, and so (iv) holds. Assume that t ≥ 4 and (iv) holds for smaller values of t. By Definition 1, G contains a pendant triangle (x, z₁, z₂) with d_G(z₁) = d_G(z₂) = 2. Since G is of Class II, d_G(x) > 4. Let $G^{\prime }=G-\left\{z_1,z_2\right\}$. Then G′ ∈ ${G}$(t − 1). If G′ is of Class II, then by induction, G′ has two pendant triangles (u₀, u₁, u₂) and (u₀, v₁, v₂) sharing a common vertex u₀. But as G is of Class II, u₀ cannot be a vertex of degree 4 in a 4-pendant triangle in G. If x ∉ {u₀, u₁, u₂, v₁, v₂}, then (u₀, u₁, u₂) and (u₀, v₁, v₂) are two pendant triangles in G, and so (iv) holds and we are done. Hence we assume that x ∉ {u₀, u₁, u₂, v₁, v₂}. If x = u₀, then (u₀, u₁, u₂), (u₀, v₁, v₂), and (u₀, z₁, z₂) are three pendant triangles with a common vertex u₀, implying (iv) also. Hence we must have x ∈ {u₁, u₂, v₁, v₂}. By symmetry, we assume that x = u₁. Then the two triangles (u₀, u₁, u₂) and (u₁, z₁, z₂) sharing a common vertex x = u₁ with d_G(u₁) = 4, and so (u₁, z₁, z₂) is a 4-pendant triangle. This implies that G is of Class I, a contradiction. Therefore, we assume that G′ is of Class I, and so G′ has a 4-pendant triangle $(u_0^{\prime },u_1^{\prime },u_2^{\prime })$ with $d_{G^{\prime }}(u_1^{\prime })=d_{G^{\prime }}(u_2^{\prime })=2$. Since G is of Class II, we may assume that $x\in \left\{u_0^{\prime },u_1^{\prime },u_2^{\prime }\right\}$. Thus either $x=u_0^{\prime }$, whence G has two pendant triangles $(x,u_1^{\prime },u_2^{\prime })$ and (x, z₁, z₂) satisfying (iv); or $x\in \left\{u_1^{\prime },u_2^{\prime }\right\}$, whence G is of Class I, contrary to the assumption that G is of Class II. □

Thus every Koch network is a 3–cactus network. However, there exist 3–cactus networks that are not Koch networks, as shown by the triangle–path $P_t^{\Delta }$ and the triangle–star $S_t^{\Delta }$ are two special classes of 3–cactus networks with t triangles. Examples of these 3–cactus networks are depicted in Figure 2.

FIGURE 2

Figure 2. The 3–cactus networks $P_t^{\Delta }$ and $S_t^{\Delta }$.

3. Algorithm for Counting the Number of Subtrees

In this section, we present a linear algorithm to count the number of subtrees in a 3–cactus network and its proof of the correctness.

Algorithm A: Initial condition: Let G_t = (V, E) be a 3–cactus network with t triangles with the vertex set V = {u₀, u₁, v₁, u₂, v₂, …, u_t, v_t} and the edge set E = {e₁, e₂, …, e_3t}, where u_i and v_i are the vertices added at the i-th iteration, for each i = 1, 2, …, t. Let η(G_t) be the number of subtrees of G_t. For every v, we assign an ordered pair of real numbers (1, 0) as the initial weight of the vertex.

Step 1. Set k = t.

Step 2. Let (a₁, b₁), (a₂, b₂), and (a₃, b₃) denote the weights of u_k, v_k and the common neighbor of u_k and v_k, respectively. Delete the vertices u_k and v_k, and reset the value of (a₃, b₃) by the following:

$\begin{array}{ll}\{\begin{array}{l}{a}_3:=a_3(3a_1{a}_2+a_1+a_2+1),\\ b_3:=b_1+b_2+b_3+a_1{a}_2+a_1+a_2.\end{array}& (1)\end{array}$

Step 3. If k = 1, then stop, output η(G_t) = a₃ + b₃. Otherwise, set k: = k − 1 and go to Step 2.

As there will be t iterations using Step 2, the complexity of Algorithm A is O(t). Since running Algorithm A to a 3–cactus networks G_t ∈ ${G}$(t) can be performed in a reverse order of building G_t as described in Definition 1, it is possible to arrange the iterations in Step 2 during the running of Algorithm A so that when Algorithm A terminates, only the initial vertex of G_t is left.

Example 1. As examples, it is inspected that η(G₀) = 1. Let G₁ = (u₀, u₁, u₂) be a triangle. Then G₁ has three subtrees avoiding u₀, namely, {u₁}, {u₂} and the complete graph on {u₁, u₂}, in G₁ that avoids the vertex u₀, and six subtrees containing u₀, induced by the vertex subset {u₀} or by the edge subsets, {u₀u₁}, {u₀u₂}, {u₀u₁, u₀u₂}, {u₀u₁, u₁u₂}, {u₀u₂, u₁u₂}. It follows that η(G₁) = 3 + 6 = 9. On the other hand, using Algorithm A, with each of u₀, u₁, u₂ assigned the initial weight (1, 0), by (1), we have a₃ = 6 and b₃ = 3. By Step 3 of Algorithm A, we also have η(G₁) = 9.

Example 2. Let H be the 3–cactus network generated by 6 iterations, as depicted in Figure 3. Using Algorithm A to H, the steps of calculating the number of subtrees of H are shown in Figure 3, which shows η(H) = 21, 096 + 378 = 21, 474.

FIGURE 3

Figure 3. An example of calculating the number of subtrees by Algorithm A, where two red vertices are deleted at each step.

Theorem 1 proves the correctness of Algorithm A.

Theorem 1. Let G_t be a 3–cactus network generated by t steps with vertex set V = {u₀, u₁, v₁, u₂, v₂, …, u_t, v_t}. Let (a, b) denote the weight of vertex u₀. Each of the following holds when Algorithm A stops.

(i) The value of a, denoted by η_u0(G_t), is the number of subtrees containing vertex u₀ in G_t.

(ii) The value of b, denoted by η(G_t − u₀), is the number of subtrees that do notcontain vertex u₀ in G_t.

(iii) The value of a + b, denoted by η(G_t), is the number of subtrees in G_t.

Proof: We argue by induction on t. If t = 1, then G₁ is a triangle. By Algorithm A, we have that a = 6 and b = 3, and by inspection, η(G₁) = 9. Thus, the result holds for t = 1. Suppose that k ≥ 2 and the result holds for all values of t < k.

The structural of G_t when t = k ≥ 2 is depicted in Figure 4, and we shall use Figure 4 and its notation to illustrate our arguments. Thus, H₁, H₂ and H₃ are the three vertex disjoint subgraphs of G_t such that H₁ contains exactly one vertex u₁ in the initial 3-cycle G₁ = (u₀, u₁, v₁), H₂ contains exactly one vertex v₁ in G₁ = (u₀, u₁, v₁) and H₃ contains exactly one vertex u₀ in G₁ = (u₀, u₁, v₁), as described in Definition 1.

FIGURE 4

Figure 4. Network G_t.

Relabel the vertices u₁, v₁, and u₀ as w₁, w₂, and w₃, when each of them is considered as a vertex in H₁, H₂, and H₃, respectively. In the rest of the proof of this theorem, we keep in mind that u₁ = w₁, v₁ = w₂, and u₀ = w₃.

As G₁ = (u₀, u₁, v₁) is an initial 3-cycle of G_t, we can view w_i as the initial vertex of the 3–cactus network H_i. For each i ∈ {1, 2, 3}, since each H_i is a 3–cactus network generated by at most t−1 iterations, by applying Algorithm A to H_i ending at the initial vertex w_i of H_i, we obtain the weights (a₁, b₁), (a₂, b₂), (a₃, b₃) of the vertices w₁, w₂, and w₃, respectively. By the induction hypothesis, we have, for i ∈ {1, 2, 3}, both (A) and (B) of the following.

$\begin{array}{l}(A){\text{a}}_{\text{i}}\text{ is the number of subtrees containing vertex}{w}_{i}in{H}_i,\\ (B){\text{b}}_{\text{i}}\text{ is the number of subtrees that do not contain vertex}\\ w_{i}in{H}_i.& (2)\end{array}$

At this stage of Algorithm A, the graph to be computed is G₁ = (u₀, u₁, v₁) with the weights assigned to u₀, u₁, v₁ being (a₃, b₃), (a₁, b₁), and (a₂, b₂), respectively. Running Algorithm A at the last iteration, we delete the vertices u₁ and v₁, and reset the weight (a, b) to the vertex u₀ by (1),

$\begin{array}{l}a=a_3(3a_1{a}_2+a_1+a_2+1)\text{ and}\\ b=b_1+b_2+b_3+a_1{a}_2+a_1+a_2.& (3)\end{array}$

Let ${T}$ denote the collection of subtrees of G_t. Define

$\begin{array}{l}{{T}}_0=\left\{T\in {T}:w_3\in V(T)\text{ and }{w}_1,w_2\notin V(T)\right\},\\ {{T}}_3=\left\{T\in {T}:w_1,w_2,w_3\in V(T)\right\},\\ {{T}}_6=\left\{T\in {T}:w_1,w_2\in V(T)\text{ and }{w}_3\notin V(T)\right\},\end{array}$

and for i ∈ {1, 2}, define

$\begin{array}{l}{{T}}_i=\left\{T\in {T}:w_3,w_i\in V(T)\text{ and }{w}_{3-i}\notin V(T)\right\},\\ {{T}}_{3+i}=\left\{T\in {T}:w_i\in V(T)\text{ and }{w}_3,w_{3-i}\notin V(T)\right\},\\ {{T}}_7=\left\{T\in {T}:w_1,w_2,w_3\notin V(T)\right\}.\end{array}$

Thus ${T}={\cup }_{i=0}^7{{T}}_i$ is a partition. By (A) and (B) in (2), $\left|{{T}}_0\right|=a_3$, $\left|{{T}}_4\right|=a_1$, $\left|{{T}}_5\right|=a_2$ and $\left|{{T}}_7\right|=b_1+b_2+b_3$. As the edge u₀u₁ lies in every tree in ${{T}}_1$, we have $\left|{{T}}_1\right|=a_1{a}_3$. Likewise, $\left|{{T}}_2\right|=a_2{a}_3$ and $\left|{{T}}_6\right|=a_1{a}_2$. Since every tree in ${{T}}_3$ contains exactly two edges in G₁ and there are three pairs of such edges in G₁, we have $\left|{{T}}_3\right|=3a_1{a}_2{a}_3$. By definition and (3),

$\begin{array}{l}{\eta }_{u_0}(G_t)={\displaystyle \sum _{i=0}^3}\left|{{T}}_i\right|=a_3+a_1{a}_3+a_2{a}_3+3a_1{a}_2{a}_3\\ =a_3(3a_1{a}_2+a_1+a_2+1)=a.\end{array}$

With a similar argument and using $\left|{{T}}_7\right|=b_1+b_2+b_3$, we have

$\begin{array}{l}\eta (G_t-u_0)={\displaystyle \sum _{i=4}^7}\left|{{T}}_i\right|=a_1+a_2+a_1{a}_2+b_1+b_2+b_3=b.\end{array}$

It follows that

$\begin{array}{l}\eta (G_t)={\displaystyle \sum _{i=0}^7}\left|{{T}}_i\right|={\eta }_{u_0}(G_t)+\eta (G_t-u_0)=a+b.\end{array}$

This shows that when Algorithm A terminates, it outputs the values as stated in Theorem 1 (i), (ii), and (iii), and justifies the correctness of Algorithm A. □

4. Upper and Lower Bounds of the Number of Subtrees in a 3–Cactus Network

The following lemma [8] will be used in our proof.

Lemma 1 ([8]). Let G be a connected network with n (n ≥ 3) vertices and let G₁ and G₂ be two subnetworks of G such that G = G₁ ∪ G₂, V(G₁) ∩ V(G₂) = {v} and E(G₁) ∩ E(G₂) = ∅. Then

$\begin{array}{r}\eta (G)=\eta (G_1)+\eta (G_2)+(\eta (G_1)-\eta (G_1-v)-1)\\ (\eta (G_2)-\eta (G_2-v)-1)-1.& (4)\end{array}$

Theorem 2. For an integer t, $S_t^{\Delta }$ is the unique 3–cactus network with the maximum number of subtrees in ${G}$(t).

Proof: We prove the theorem by induction on t, and observe that the theorem holds when t ∈ {1, 2}. Let k ≥ 3 be an integer and assume that the result is true for t < k. We consider that case when t = k.

Let G′ be a 3–cactus network with k triangles, and let (u, v, w) be a pendant triangle such that $d_{G^{\prime }}(u)\ge 4$ and $d_{G^{\prime }}(w)=d_{G^{\prime }}(v)=2$. By Observation 1(iii), u is a cut vertex. Let $G_1^{\prime }=G^{\prime }-\left\{w,v\right\}$ and $G_2^{\prime }=(u,v,w)$.

By the induction hypothesis, $\eta (G_1^{\prime })\le \eta (S_{k-1}^{\Delta })$, where the equality holds if and only if $G_1^{\prime }=S_{k-1}^{\Delta }$. If $G^{\prime }=S_k^{\Delta }$, then $G_1^{\prime }-u=(k-1)P_2$, where P₂ is the path with 2 vertices. If $G^{\prime }\ne S_k^{\Delta }$, then (k − 1)P₂ must be a proper subnetwork of $G_1^{\prime }-u$, which implies $\eta (G_1^{\prime }-u)>\eta ((k-1)P_2)$. Since $G_2^{\prime }=(u,v,w)$ and $G_2^{\prime }-u=P_2$, it follows from (4) that $\eta (G^{\prime })\le \eta (S_k^{\Delta })$, where the equality holds if and only if $G^{\prime }=S_k^{\Delta }$, which completes the proof. □

The next lemma presents some observations which follow from Algorithm A.

Lemma 2. Let G_t ∈ ${G}$(t) be a 3–cactus network with a pendant triangle (u, v, w) such that d_Gt(w) = d_Gt(v) = 2, and let G_t−1 = G_t − {w, v}. Then

(i) η(G_t) = 5η_u(G_t−1) + η(G_t−1) + 3;

(ii) η_w(G_t) = 4η_u(G_t−1) + 2. □

Lemma 3. Let t ≥ 3 be an integer and G ∈ ${G}$(t) be a Class II 3–cactus network. Then there exists a Class I 3–cactus network G′ in ${G}$(t) satisfying η(G′) < η(G).

Proof: Let G ∈ ${G}$(t) be a Class II 3–cactus network. By Observation 1(iv), G contains two pendant triangles (u₀, u₁, u₂) and (u₀, u, w) such that d_G(u₀) ≥ 6 and d_G(u₁) = d_G(u₂) = d_G(u) = d_G(w) = 2, as depicted in Figure 5A. Let H be the graph depicted in Figure 5. Denote by a₀ (b₀, respectively) the number of subtrees containing u₀ (not containing u₀) in H. By Algorithm A, when deleting the vertices u₁ and u₂ from G, the weight of vertex u₀, denoted by (a_u0, b_u0), can be reset as follows: a_u0 = 6a₀ and b_u0 = b₀ + 3. Then in the step of deleting the vertices u₀ and w from G, the weight of vertex u is given by η_u(G) = 4a_u0 + 2, η(G − u) = 2a_u0 + b_u0 + 1. Thus, we have η(G) = 36a₀ + b₀ + 6.

FIGURE 5

Figure 5. (A) The network G. (B) The network G′.

Let G′ denote the graph depicted in Figure 5B, which is obtained from H by identifying vertex u₀ of H and a vertex of degree 2 in $P_2^{\Delta }$. By Algorithm A, when deleting the vertices u₀ and u₂ from G′, the weight of vertex u₁, denoted by (a_u1, b_u1), is obtained as a_u1 = 4a₀ + 2, b_u1 = 2a₀ + b₀ + 1. Then in the step of deleting the vertices u₁ and w from G′, the weight of vertex u is given by ${\eta }_u(G^{\prime })=4a_{u_1}+2$, $\eta (G^{\prime }-u)=2a_{u_1}+b_{u_1}+1$. Thus, we have $\eta (G^{\prime })=26a_0+b_0+16$. Algebraic computation yields that $\eta (G)-\eta (G^{\prime })=10a_0-10>0$, and so the lemma is proved. □

Theorem 3. For an integer t, $P_t^{\Delta }$ is the unique 3–cactus network with the minimum number of subtrees in ${G}$(t).

Proof: By Lemma 3, within the family ${G}$(t), for every 3–cactus network G in Class II, there is always a 3–cactus network G′ in Class I containing fewer subtrees. Thus, we only need to show that $P_t^{\Delta }$ is the unique 3–cactus network with the minimum number of subtrees, among all the networks of ${G}$(t) in Class I.

We prove the conclusion by induction on t. Let G_t ∈ ${G}$(t) be a 3–cactus network with a pendant triangle (u, v, w) such that d_Gt(u) = 4 and d_Gt(w) = d_Gt(v) = 2. We will prove the following results: (i) $P_t^{\Delta }$ is the unique 3–cactus network with the minimum number of subtrees in ${G}$(t); (ii) η_w(G_t) attains the minimum value if and only if $G_t=P_t^{\Delta }$.

The results hold for t = 1 and 2. Suppose that k ≥ 3 and the results are true for t < k, we consider the case when t = k. Let G_k−1 = G_k − {w, v}. Since d_Gk−1(u) = 2, by Lemma 2 and the induction hypothesis, we have that

$\begin{array}{l}{\eta }_w(G_k)=4{\eta }_u(G_{k-1})+2\ge 4{\eta }_u(P_{k-1}^{\Delta })+2\end{array}$

and

$\begin{array}{l}\eta (G_k)=5{\eta }_u(G_{k-1})+\eta (G_{k-1})+3\ge 5{\eta }_u(P_{k-1}^{\Delta })+\eta (P_{k-1}^{\Delta })+3,\end{array}$

where the equalities hold if and only if $G_k=P_k^{\Delta }$. □

With Algorithm A, we can compute the numbers of subtrees of $S_t^{\Delta }$ and $P_t^{\Delta }$, respectively. Let u₀ be the center of $S_t^{\Delta }$ and v₀ be a vertex of degree 2 in a pendant triangle of $P_t^{\Delta }$. From Algorithm A, it follows that

$\begin{array}{l}\{\begin{array}{cc}{\eta }_{u_0}(S_t^{\Delta })=6{\eta }_{u_0}(S_{t-1}^{\Delta }),& {\eta }_{u_0}(S_0^{\Delta })=1,\\ \eta (S_t^{\Delta }-u_0)=\eta (S_{t-1}^{\Delta }-u_0)+3,& \eta (S_0^{\Delta }-u_0)=0,\end{array}\end{array}$

and

$\begin{array}{l}\{\begin{array}{ll}{\eta }_{v_0}(P_t^{\Delta })=4{\eta }_{v_0}(P_{t-1}^{\Delta })+2,& {\eta }_{v_0}(P_0^{\Delta })=1,\\ \eta (P_t^{\Delta }-v_0)=\eta (P_{t-1}^{\Delta }-v_0)+2{\eta }_{v_0}(P_{t-1}^{\Delta })+1,& \eta (P_0^{\Delta }-v_0)=0.\end{array}\end{array}$

By algebraic manipulations, we obtain that

$\begin{array}{ll}\eta (S_t^{\Delta })=6^t+3t& (5)\end{array}$

and

$\begin{array}{ll}\eta (P_t^{\Delta })=\frac{25}{9}\times 4^t-\frac{t}{3}-\frac{16}{9}.& (6)\end{array}$

By Theorems 2 and 3, $\eta (S_t^{\Delta })$ and $\eta (P_t^{\Delta })$ represent the upper and lower bounds of the number of subtrees in a 3–cactus network with t triangles, respectively. As shown in (5) and (6), the numbers of subtrees in $S_t^{\Delta }$ and $P_t^{\Delta }$ grow exponentially as the number of triangles increases, as seen in Figure 6, where the curves of values of $\eta (S_t^{\Delta })$ and $\eta (P_t^{\Delta })$ vs. t are plotted.

FIGURE 6

Figure 6. Log-log plot of $\eta (P_t^{\Delta })$ and $\eta (S_t^{\Delta })$ vs. t.

5. The Entropy of Subtrees of Koch Networks

The entropy of spanning trees has been well-studied, as seen in Lyons et al. [21] and Zhang et al. [22, 23], among others. The entropy of subtrees can be similarly defined.

Definition 2. Let G be a network with N(G) vertices. The entropy of subtrees of G is defined as follows:

$\begin{array}{l}E(G)={\displaystyle \underset{N(G)\to \infty }{lim}}\frac{ln\eta (G)}{N(G)}.\end{array}$

Let T be a tree with n vertices. Székely and Wang [2] and Yan and Yeh[3] obtained the results of the number of subtrees in a tree as follows: $\frac{n(n+1)}{2}=\eta (P_n)\le \eta (T)\le \eta (K_{1,n-1})=2^{n-1}+n-1$, where P_n and K_1,n−1 are the path and star with n vertices, respectively. So, we have

$\begin{array}{l}E(K_{1,n-1})={\displaystyle \underset{n\to \infty }{lim}}\frac{ln(2^{n-1}+n-1)}{n}=ln2\end{array}$

and

$\begin{array}{l}E(P_n)={\displaystyle \underset{n\to \infty }{lim}}\frac{ln\frac{n(n+1)}{2}}{n}=0.\end{array}$

From Equations (5) and (6), we can calculate the entropies of subtrees of $S_t^{\Delta }$ and $P_t^{\Delta }$, respectively, as follows:

$\begin{array}{l}E(S_t^{\Delta })={\displaystyle \underset{t\to \infty }{lim}}\frac{ln(6^t+3t)}{2t+1}=\frac{ln6}{2}\end{array}$

and

$\begin{array}{l}E(P_t^{\Delta })={\displaystyle \underset{t\to \infty }{lim}}\frac{ln(\frac{25}{9}\times 4^t-\frac{t}{3}-\frac{16}{9})}{2t+1}=ln2.\end{array}$

Therefore, for a tree T with n vertices and a 3–cactus network G_t with t triangles, we have

$\begin{array}{l}E(P_n)\le E(T)\le E(K_{1,n-1})=E(P_t^{\Delta })\le E(G_t)\le E(S_t^{\Delta }).\end{array}$

Now we calculate the entropy of subtrees of Koch networks. The following definition and notation will be used in this section.

Let α(x) and β(x) be two infinities when x is close to infinity. If $\underset{x\to \infty }{\lim }\frac{\alpha (x)}{\beta (x)}=1$, we say that α(x) and β(x) are equivalent infinities, and write α(x) ~ β(x) (x → ∞).

For Koch networks, another construction is given as follows. The network G_{m, g} is obtained from G_m,i (i = 0, 1, 2, …, g − 1) and the initial triangle by adding m copies of G_m,i to each of the three hub vertices of the initial triangle, as shown in Figure 7.

FIGURE 7

Figure 7. Another construction of Koch networks. The network after g iterations, G_m,g, consists of 3m copies of G_m,i labeled by $G_{m,i}^j(j=1,2,\dots ,3m;i=0,1,2,\dots ,g-1)$.

According to the construction, G_m,g has the form illustrated in Figure 8, where H_g is the subnetwork of G_m,g obtained from m copies of G_m,i (i = 0, 1, 2, …, g − 1) by identifying mg hub vertices X in m copies of G_m,i (i = 0, 1, 2, …, g − 1). Then η_X(H_g) and η(H_g − X) are given by

$\begin{array}{ll}\{\begin{array}{c}{\eta }_X(H_g)=({\displaystyle \prod _{i=0}^{g-1}}{\eta }_X(G_{m,i}){)}^m,\\ \eta (H_g-X)=m{\displaystyle \sum _{i=0}^{g-1}}\eta (G_{m,i}-X).\end{array}& (7)\end{array}$

FIGURE 8

Figure 8. Koch network G_m,g.

By Algorithm A to the initial triangle (X, Y, Z), we get that

$\begin{array}{ll}\{\begin{array}{c}{\eta }_X(G_{m,g})={\eta }_X(H_g)\left[3{({\eta }_X(H_g))}^2+2{\eta }_X(H_g)+1\right],\\ \eta (G_{m,g}-X)=3\eta (H_g-X)+{({\eta }_X(H_g))}^2+2{\eta }_X(H_g)\end{array}& (8)\end{array}$

and initial conditions are η_X(G_m,0) = 6 and η(G_m,0 − X) = 3. So, we arrive at

$\begin{array}{ll}\eta (G_{m,g})=3{\eta }_X(H_g)\left[{({\eta }_X(H_g))}^2+{\eta }_X(H_g)+1\right]+3\eta (H_g-X).& (9)\end{array}$

From Equation (7), we obtain

$\begin{array}{ll}\{\begin{array}{c}{\eta }_X(H_g)={\eta }_X(H_{g-1}){({\eta }_X(G_{m,g-1}))}^m,\\ \eta (H_g-X)=\eta (H_{g-1}-X)+m\eta (G_{m,g-1}-X).\end{array}& (10)\end{array}$

According to the above theoretical analysis, we give a linear algorithm for counting the number of subtrees in a Koch network, as follows.

Algorithm B: Initial condition: Let G_m,g be a Koch network generated by g iterations and X, Y and Z be three hub vertices in G_m,g. Let η(G_m,g) denote the number of subtrees of G_m,g.

Step 1. Take k = 1, a = 6, b = 3, x = 1 and y = 0.

Step 2. Suppose that, at iteration k, the number of subtrees containing X in H_k and the number of subtrees not containing X in H_k are denoted by x and y, respectively. One gets the values x and y as

$\begin{array}{l}\{\begin{array}{c}x:=x{(a)}^m,\\ y:=y+mb.\end{array}\end{array}$

Step 3. Suppose that a and b denote the number of subtrees containing X in G_m,k and the number of subtrees not containing X in G_m,k, respectively. We get a and b by

$\begin{array}{l}\{\begin{array}{c}a:=x(3x^2+2x+1),\\ b:=3y+x^2+2x.\end{array}\end{array}$

Step 4. If k = g, then stop, output η(G_m,g) = a + b. Otherwise, k: = k + 1, go to Step 2.

It is not difficult to see that the complexity of the above algorithm is O(g). In order to calculate the entropy of subtrees of Koch networks, we need the following lemma:

Lemma 4. Let G_m,g be a Koch network generated by g iterations. Then

$\begin{array}{l}\eta (G_{m,g})~3^{{(3m+1)}^{g-1}}\times 6^{3m{(3m+1)}^{g-1}}(g\to \infty ).\end{array}$

Proof: From Equation (10), we have the following recurrence relations:

$\begin{array}{l}\{\begin{array}{l}{\eta }_X(H_g)={\eta }_X(H_{g-1}){\left[{\eta }_X(H_{g-1})(3{({\eta }_X(H_{g-1}))}^2+2{\eta }_X(H_{g-1})+1)\right]}^m,\\ \eta (H_g-X)=(3m+1)\eta (H_{g-1}-X)+m\left[{({\eta }_X(H_{g-1}))}^2+2{\eta }_X(H_{g-1})\right].\end{array}\end{array}$

It is easy to obtain ${\eta }_X(H_g)~3^m\times {({\eta }_X(H_{g-1}))}^{3m+1}(g\to \infty )$. Note that ${\eta }_X(H_1)=6^m$.

It follows that

$\begin{array}{ll}{\eta }_X(H_g)~3^{\frac{1}{3}\left[{(3m+1)}^{g-1}-1\right]}\times 6^{m{(3m+1)}^{g-1}}(g\to \infty ).& (11)\end{array}$

Since η(H₁ − X) = 3m, we have

$\begin{array}{l}\eta (H_g-X)=3m{(3m+1)}^{{}^{g-1}}+m{\displaystyle \sum _{i=1}^{g-1}{(3m+1)}^{{}^{g-1-i}}}[{({\eta }_X(H_i))}^2\\ +2{\eta }_X(H_i)]\\ \le 3m{(3m+1)}^{g-1}+m(g-1){(3m+1)}^{g-2}\\ [{({\eta }_X(H_{g-1}))}^2+2{\eta }_X(H_{g-1})]\\ ~m(g-1){(3m+1)}^{g-2}{({\eta }_X(H_{g-1}))}^2(g\to \infty ).& (12)\end{array}$

From Equations (11) and (12), we get that $\underset{x\to \infty }{\lim }\frac{\eta (H_g-X)}{{({\eta }_X(H_g))}^3}=0$. From Equation (9), it is easy to see that

$\begin{array}{ll}\eta (G_{m,g})~3{({\eta }_X(H_g))}^3(g\to \infty ).& (13)\end{array}$

From Equations (11) and (13), we conclude that

$\begin{array}{l}\eta (G_{m,g})~3^{{(3m+1)}^{g-1}}\times 6^{3m{(3m+1)}^{g-1}}(g\to \infty ).\end{array}$

The proof is thus complete. □

From Lemma 4, we can calculate the entropy of subtrees of Koch networks, as

$\begin{array}{l}E(G_{m,g})={\displaystyle \underset{g\to \infty }{lim}}\frac{ln\eta (G_{m,g})}{2{(3m+1)}^g+1}\\ ={\displaystyle \underset{g\to \infty }{lim}}\frac{ln[3^{{(3m+1)}^{g-1}}\times 6^{3m{(3m+1)}^{g-1}}]}{2{(3m+1)}^g+1}=\frac{ln6}{2}-\frac{ln2}{2(3m+1)}.\end{array}$

Next, we can compare the entropy of subtrees of Koch networks with those of $P_t^{\Delta }$ and $S_t^{\Delta }$. Observe that $ln2<\frac{ln6}{2}-\frac{ln2}{2(3m+1)}<\frac{ln6}{2}$ and $\underset{m\to \infty }{\lim }E(G_{m,g})=\underset{m\to \infty }{\lim }(\frac{ln6}{2}-\frac{ln2}{2(3m+1)})=\frac{ln6}{2}.$ Thus, the following inequality holds.

$\begin{array}{l}E(P_t^{\Delta })<E(G_{m,g})<E(S_t^{\Delta }).\end{array}$

The entropy of subtrees of Koch networks reflects the fact that although the number of subtrees of Koch networks grows exponentially, the growth rate is lower than that of the triangle–star and higher than that of the triangle–path. This result suggests that if the vertex failure probability p satisfies p → 0, the edge failure probability q satisfies q → 1, and p and 1 − q are equivalent infinitesimals, then Koch networks are less reliable than the triangle–star, and more reliable than the triangle–path. Noticing that E(G_m,g) can be considered as a function of the variable m, it is easy to see that the entropy of subtrees of Koch networks tends to the maximum entropy of subtrees of all 3–cactus networks as m tends to infinity.

6. Conclusions

In this paper, we have given the definition of the entropy of subtrees, which is used to compare the average number of subtrees for networks of different sizes. We have established a linear algorithm for counting the number of subtrees in any 3–cactus network, and characterized the 3–cactus networks with upper and lower bounds of the entropy of subtrees among all 3–cactus networks with t triangles. In order to avoid exponential computation, we have also proposed a linear algorithm for calculating the number of subtrees in Koch networks. Finally, we have determined the entropy of subtrees of Koch networks which tends to the maximum entropy of subtrees of all 3–cactus networks as m tends to infinity.

Data Availability Statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

Author Contributions

LD, HZ, and H-JL contributed the conception and design of the study. LD organized the literature, wrote the simulation programs, and performed the design of figures. LD and HZ wrote the first draft of the manuscript. All authors contributed to the manuscript revision, read, and approved the submitted version.

Funding

This work was supported by National Science Foundation of China (Grant No. 11661069), the Program for Changjiang Scholars and Innovative Research Team in Universities (Grant No. IRT-15R40), the Key Laboratory of Qinghai Province (Grant No. 2013-Z-Y17), the Research Fund for the Chunhui Program of Ministry of Education of China, the Nature Science Foundation from Qinghai Province (Grant No. 2018-ZJ-718), and the Research Foundation from Qinghai Normal University (Grant No. 2018zr010).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors were very grateful to the referees and the editor for their valuable comments and suggestions, which significantly improved the presentation of this manuscript.

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