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Enumeration of spanning trees in the sequence of Dürer graphs

  • Shixing Li EMAIL logo
Published/Copyright: December 29, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper, we calculate the number of spanning trees in the sequence of Dürer graphs with a special feature that it has two alternate states. Using the electrically equivalent transformations, we obtain the weights of corresponding equivalent graphs and further derive relationships for spanning trees between the Dürer graphs and transformed graphs. By algebraic calculations, we obtain a closed-form formula for the number of spanning trees with regard to iteration step. Finally we compare the entropy of our graph with other studied graphs and see that its value of entropy lies in the interval of those of graphs with average degree being 3 and 4.

Keywords: Spanning trees; Electrically equivalent transformation; Entropy
MSC 2010: 05C30; 05C50; 05C63

1 Introduction

In the study of networks, spanning trees are related to many aspects of networks, such as reliability [1, 2], consensus [3], random walks [4] and nonlinear dynamics [5, 6]. A spanning tree of a connected graph is defined as a minimal set of edges that connect all its nodes. On the other hand, enumeration of spanning trees has been applied in mathematics [7], computer science [8, 9], physics [10, 11], and chemistry [12], to name just a few.

Recently, counting the number of spanning trees has attracted increasing attention [13, 14, 15, 16]. It is known that the number of spanning trees can be obtained by matrix-tree theorem [13]. Due to the complexity and diversity of networks, analytically enumerating spanning trees is challenging. For fractal networks, the existing works [17, 18, 19, 20] applied the self-similarity, spanning forests, and Laplacian spectrum to obtain exact formulae of spanning trees, while for some self-similar graphs, e.g., generalized Petersen graphs, those methods are not effective. Recently, the enumeration of spanning trees has been investigated by the electrically equivalent transformations in Refs. [21, 22, 23].

A sequence of Dürer graphs is the skeleton of Dürer’s solid, which belongs to generalized Petersen graphs. This graph has a special feature that it has two alternate states. Calculating the number of spanning trees in this family of graph by its laplacian spectrum does not work. To the best of our knowledge, few analytical results involve the derivation of enumeration of spanning trees. Here we employ the knowledge of electrical networks, where an edge-weighted graph is regarded as an electrical network with the weights equalling the conductances of the corresponding edges. Using the electrically equivalent transformations provided in Refs. [24, 25], we obtain some relationships for spanning trees between the original graph and transformed graphs. We then obtain a closed-form formula for enumeration of spanning trees. Finally we calculate the entropy of spanning trees and compare it with those of other studied graphs.

In Section 2, the construction of the sequence of Dürer graphs is presented. Section 3 shows the electrically equivalent transformations of graphs. The detailed calculations of spanning trees are provided in Section 4. Finally, the conclusions are included in the last section.

2 Model presentation

The sequence of Dürer graphs is denoted by Gn(n ≥ 1) after n steps, which is constructed as follows:

For n = 1, G1 is a triangle.

For n = 2, G2 is obtained from G1, where a hexagon is put into the triangle and three new adjacent nodes in the hexagon are connected to the nodes in the triangle.

For n ≥ 3, Gn is produced by Gn−1. If n is an even number, a hexagon is inserted into Gn−1; if n is an odd number, a triangle is embedded into Gn−1. Figure 1 shows this construction process.

Figure 1 The Dürer graphs Gn produced at steps n = 1, 2, 3 and 4.
Figure 1

The Dürer graphs Gn produced at steps n = 1, 2, 3 and 4.

According to the construction, the number of total vertices Vn and edges En are,

Vn=9n2−32,En=15n2−92,n=1,3,⋯,

and

Vn=9n2,En=15n2−3,n=2,4,⋯.

Then the average degree is 〈k〉n = 2EnVn, which is approximately 103 for a large t.

3 Electrically equivalent transformations

According to the structures of Gn, the final state is either a triangle or a hexagon. When the final state is a hexagon, using the star-triangle electrically equivalent transformation, we obtain a new graph, denoted by Ft and the weights in the new triangles become 13 , other edge weights remain unchanged. Figure 2 shows the transformation between G2 and F1. Using the results in Ref. [24], we transform a star graph with weights a, b and c to a triangle graph H*, where the weights are bca+b+c,aca+b+c and aba+b+c. Then we obtain the relationship of weighted spanning trees between H and H*, i.e., τ(H) = (a + b + c)τ(H*). Since the graph Gn contains 32 n identical star graphs, we obtain

τ(Gn)=332nτ(Ft). (1)
Figure 2 The transformation from G2 to F1.
Figure 2

The transformation from G2 to F1.

In the same way, when the final state is a triangle, we transform Gn into Ft∗ . Figure 3 gives the transformation between G3 and F1∗ . The weighted spanning trees reads as

τ(Gn)=332(n−1)τ(Ft∗). (2)
Figure 3 The transformation from G3 to F1∗ $F_{1}^{*} $.
Figure 3

The transformation from G3 to F1∗ .

In order to calculate the functions τ(Ft) and τ( Ft∗ ), we provide the following two Lemmas and three corollaries.

Lemma 3.1

For the edge-weighted graph F1 (see Fig. 4), suppose the weights of the innermost triangle, outermost triangle and the linked edges are a, c and b, respectively. Then, we have

τ(F1)=6b(2ab+2bc+3ac+b2)2.
Figure 4 The transformations from F1 to X5.
Figure 4

The transformations from F1 to X5.

Proof

We firstly transform the innermost triangle with weights a into an electrically equivalent star graph denoted by X1 with weights 3a, then τ(X1) = 9aτ(F1).

Using the star-triangle transformation, three star graphs are changed into three connected curved edge triangles, where the weights of innermost six curved edges are 3ab3a+2b, and the weights of outer three curved edges are b23a+2b. The transformed graph is denoted by X2, then τ(X2) = (13a+2b)3 τ(X1).

Merging six pairs of parallel edges into single edges produces a new graph denoted by X3, which includes a triangle with weights c + b23a+2b and a star graph with weights 6ab3a+2b. Then τ(X3) = τ(X2). Using the star-triangle transformation again, we obtain a curved edge triangle with weights 2ab3a+2b, then τ(X4) = 3a+2b18ab τ(X3).

Finally, merging two parallel edges into a single edge to form a new triangle with weights c + b2+2ab3a+2b, we obtain τ(X5) = τ(X4). Through the above five transformations, we obtain τ(F1) = 6b(2ab + 2bc + 3ac + b2)2. The whole electrically equivalent transformations from F1 to X5 are shown in Fig. 4.□

Lemma 3.2

For the edge-weighted graph Y1(see Fig. 5), the weights of the innermost triangle, outermost triangle and the linked edges are x, z and y. Then,

τ(Y1)=3y(xy+3xz+yz)2.

Proof

Using the same transformation between F1 and X1, we change Y1 into Y2, then τ(Y2) = 9xτ(Y1). Merging two serial edges with the weights 3x and y into a single edge with the weight 3xy3x+y, we obtain τ(Y3) = 1(3x+y)3 τ(Y2). Then implementing the star-triangle transformation and the curved edge triangle with weights xy3x+y. Thus, τ(Y4)=3x+y9xyτ(Y3).

Finally merging parallel edges into a single edge with weight z + xy3x+y gives τ(Y5) = τ(Y4). Combining the above-mentioned transformation, we obtain τ(Y1) = 3y(xy + 3xz + yz)2. Figure 5 gives the electrically equivalent transformations between Y1 and Y5. According to Lemmas 3.1 and 3.2, we obtain the following corollaries.

Figure 5 The transformations from Y1 to Y5.
Figure 5

The transformations from Y1 to Y5.

□

Corollary 3.3

Considering a new graph F2 formed by connecting two graphs F1 through three edges with the weight y, we obtain

τ(F2)=2by(2b+3a)2(3x+y)2τ(F1),

where x = c + b2+2ab3a+2b.

Proof

Based on the transformations in Fig. 6, we have τ(F2) = 2b(2b + 3a)2τ( F1∗ ) and τ( F1∗ ) = y(3x + y)2τ(F1).

Figure 6 The transformations from F2 to F1.
Figure 6

The transformations from F2 to F1.

□

Corollary 3.4

Based on the transformations of F2, we obtain a relationship of enumerating spanning trees between Ft and F1, i.e.,

τ(Ft)=(23)t−1∏i=2t(3+14ri)2⋅τ(F1), (3)

where ri = 16+75ri+127+126ri+1, and ri is the edge weights in the innermost triangle of Fi.

Proof

According to the transformations (F1 → X5 and Y1 → Y5) in Figs. 4 and 6, we obtain x = c+b2+2ab3a+2b and r1 = z + xy3x+y. Let z=b=13, c = y = 1, a = r2, it gives x = 33r2+727r2+6 and r1=16+75r227+126r2. Further, τ(F2) = 23 (3 + 14r2)2 ⋅ τ(F1). Through the transformations between Ft and F1 and by induction, Equation (3) holds.□

Corollary 3.5

The graphs Ft∗ are produced by inserting a triangle with the weights 1 into Ft, and connecting them by linked-edges with the weights 1. Then,

τ(Ft∗)=16τ(Ft). (4)

Proof

Let x = y = 1, z = 13 and from Lemma 3.2, it gives τ(Y1) = 16τ(Y5). Using the transformations from Ft∗ to Ft, Corollary 3.5 is established.□

4 Calculating the number of spanning trees

Using the expression rt−1=16+75rt27+126rt and denoting the coefficients of 27 + 126rt and 16 + 75rt as At and Bt, we obtain

3+14rt=A0(27+126rt)+B0(16+75rt),3+14rt−1=A1(27+126rt)+B1(16+75rt)9[A0(27+126rt)+B0(16+75rt],⋮⋮3+14rt−i=Ai(27+126rt)+Bi(16+75rt)9[Ai−1(27+126rt)+Bi−1(16+75rt], (5)

3+14rt−(i+1)=Ai+1(27+126rt)+Bi+1(16+75rt)9[Ai(27+126rt)+Bi(16+75rt],⋮⋮3+14r2=At−2(27+126rt)+Bt−2(16+75rt)9[At−3(27+126rt)+Bt−3(16+75rt)], (6)

where A0 = 19 , B0 = 0; A1 = 3, B1 = 14. Substituting Eq. (5) into Eq. (3) yields

τ(Ft)=2t−1⋅3−5t+9[At−2(27+126rt)+Bt−2(16+75rt)]2⋅τ(F1). (7)

Let a = r1, b = 13 , c = 1 and from Lemma 3.1, we obtain

τ(F1)=281⋅(7+33r1)2. (8)

By the relationship between rt and rt−1 and Eqs. (5) and (6), we obtain

At+1=102At−9At−1;Bt+1=102Bt−9Bt−1.

Their characteristic equation is

λ2−102λ+9=0,

with two roots being λ1=51+362 and λ2=51−362. Then the general solutions are

At=a1λ1t+a2λ2t;Bt=b1λ1t+b2λ2t.

Using the initial conditions A0=19,B0=0;A1=3,B1=14 gives

At=3−254λ1t+3+254λ2t;Bt=7272λ1t−7272λ2t. (9)

In the sequel, we calculate the values of r1. By rt−1=16+75rt27+126rt, its characteristic equation is 63x2−24x−8 = 0, which has two roots x1=4+6221 and x2=4−6221.

Subtracting these two roots from both sides of rt−1=16+75rt27+126rt yields

rt−1−4+6221=16+75rt27+126rt−4+6221=51−36227+126rt(rt−4+6221),rt−1−4−6221=16+75rt27+126rt−4−6221=51+36227+126rt(rt−4−6221).

Let at=rt−4+6221rt−4−6221, then,

at−1=17−12217+122at,

where

a1=r1−4+6221r1−4−6221=(17−12217+122)t−1at.

Hence, the expression of r1 reads as

r1=(62−4)(577−4082)t−1at+4+6221−21(577−4082)t−1at. (10)

If rt=13, then at=3−623+62. Plugging Eqs. (7)-(10) into Eq. (1) gives

τ(Gn)=2n2⋅3−n+5(92+65224λ1n−42+92−65224λ2n−42)2(7+33r1)2,

where r1=(22−4)(577−4082)n2−1+22+4(62−3)(577−4082)n2−1+62+3,λ1=51+362 and λ2=51−362.

If rt=712, then at=11−8211+82. Inserting Eqs. (4) and (7)-(10) into Eq. (2) yields

τ(Gn)=2n+72⋅3−n+6(536+379296λ1n−52+536−379296λ2n−52)2(7+33r1)2,

where r1=(142−20)(577−4082)n−32+142+20(242−33)(577−4082)n−32+242+33. Then, we have the following theorem.

Theorem 4.1

The enumeration of spanning trees in the sequence of Dürer graphs is as follows:

τ(Gn)=2n+72⋅36−n(536+379296λ1n−52+536−379296λ2n−52)2⋅Φ1,n=1,3,⋯,2n2⋅35−n(92+65224λ1n−42+92−65224λ2n−42)2⋅Φ2,n=2,4,⋯,

where Φ1 = (7+33r1)2 with r1=(142−20)(577−4082)n−32+142+20(242−33)(577−4082)n−32+242+33 and Φ2 = (7+33r1)2 with r1 = (22−4)(577−4082)n2−1+22+4(62−3)(577−4082)n2−1+62+3.

5 Entropy of spanning trees

Using the obtained results for enumeration of spanning trees, we calculate the entropy of spanning trees, denoted by E(G), which is given by,

E(G)=limn→∞ln⁡τ(Gn)Vn,=ln⁡2−2ln⁡3+2ln⁡(51+362)9≈0.860.

Now we compare the value of entropy in our graph with other graphs. For the graphs with average degree 3, the entropy of infinite outerplanar small-world graphs [26] is 0.657, the values of entropy in 3-12-12 and 4-8-8 lattices [27] are 0.721 and 0.787, and the honeycomb lattice [28] is 0.807. While for the graphs with average degree 4, the entropy of the pseudofractal fractal web [29] is 0.896, the fractal scale-free lattice [20] is 1.040, the values of the two-dimensional Sierpinski gasket [15] and the square lattice [28] are 1.049 and 1.166. The entropy of spanning trees in our graph is 0.860, which is larger than those of graphs with average degree 3, but smaller than those of graphs with average degree 4.

6 Conclusions

In the present study, we have used the electrically equivalent transformations to solve the number of spanning trees in the sequence of Dürer graphs. Compared to existing methods on enumeration of spanning trees, this method is effective and simple. Applying the transformations, we have converted this graph into a triangle, and obtained the relationships of corresponding edge weights. Using the obtained method, we could calculate the spanning trees of Dürer-like graphs, e.g., the cylinders width being an even number. In addition, our results have shown that the entropy is related to the average degree, whether this conclusion holds for other graphs needs further study.

Acknowledgement

This work was supported by the Planning Project of Social Science of Zhejiang Province (Grant No. 18NDJC138YB) and First Class Discipline of Zhejiang-A (Zhejiang University of Finance and Economics-Statistics).

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Received: 2016-10-12
Accepted: 2017-12-7
Published Online: 2017-12-29

© 2017 Li

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2017-0136
Keywords for this article
Spanning trees; Electrically equivalent transformation; Entropy
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BY-NC-ND 4.0

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