The normalized Laplacian spectrum of subdivisions of a graph

The normalized Laplacian spectrum of subdivisions of a graph

Pinchen Xie Zhongzhi Zhang zhangzz@fudan.edu.cn Francesc Comellas comellas@ma4.upc.edu Department of Physics, Fudan University, Shanghai 200433, China School of Computer Science, Fudan University, Shanghai 200433, China Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China Department of Applied Mathematics IV, Universitat Politècnica de Catalunya, 08034 Barcelona Catalonia, Spain
Abstract

Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny’s constant and number of spanning trees.

keywords:
Normalized Laplacian spectrum, Subdivision graph, Degree-Kirchhoff index, Kemeny’s constant, Spanning trees
journal: Theoretical Computer Science

1 Introduction

Spectral analysis of graphs has been the subject of considerable research effort in theoretical computer science Dasgupta et al. (2004); Spielman (2007); Tran et al. (2013), due to its wide applications in this area and in general Cvetković and Simić (2011); Arsić et al. (2012). In the last few decades a large body of scientific literature has established that important structural and dynamical properties of networked systems are encoded in the eigenvalues and eigenvectors of some matrices associated to their graph representations. The spectra of the adjacency, Laplacian and normalized Laplacian matrices of a graph provide information on bounds on the diameter, maximum and minimum degrees, possible partitions, and can be used to count the number of paths of a given length, number of triangles, total number of links, number of spanning trees and many more invariants. Dynamic aspects of a network, like its synchronizability and random walks properties can be determined from the eigenvalues of the Laplacian and normalized Laplacian matrices which allow also the calculation of some interesting graph invariants like the Kirchhoff index Godsil and Royle (2001); Chung (1997); Brouwer and Haemers (2012).

We notice that in the last years there has been an increasing interest in the study of the normalized Laplacian as many measures for random walks on a network are linked to the eigenvalues and eigenvectors of normalized Laplacian of the associated graph, including the hitting time, mixing time and Kemeny’s constant which can be used as a measure of efficiency of navigation on the network, see Lovász (1993); Zhang et al. (2013); Kemeny and Snell (1976); Levene and Loizou (2002).

However, the normalized and standard Laplacian matrices of a network behave quite differently Chen et al. (2004), and even if the spectrum of one matrix can be determined this does not mean that the other can also be evaluated if the graphs are not regular. As an example, the eigenvalues of the Laplacian of Vicsek fractals can be found analytically Blumen et al. (2003), but until now it has not been possible to obtain the spectra of their normalized Laplacian. Thus, the spectra of the standard and normalized Laplacian matrices must be considered independently.

In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs and we use these results to find the values of their multiplicative degree-Kirchhoff index, Kemeny’s constant and number of spanning trees.

2 Preliminaries

Let be any simple connected graph with vertex set and edge set . Let denote the number of vertices of and its number of edges.

Definition

(Shirai Shirai (1999)) The subdivision graph of , denoted by , is the graph obtained from by inserting an additional vertex to every edge of .

We denote . The -th subdivision of is obtained through the iteration and and denote the total number of vertices and edges of . Figure 1 illustrates the iterated subdivisions of four-vertex complete graph .

From the definition of the subdivision graph, it is obvious that and . Thus, for , we have

(1)

Moreover, for any vertex, once it is created, its degree remains unchanged as grows.

Figure 1: , and . Black vertices denote the initial vertices of while red and green vertices are those introduced to obtain and , respectively.
Definition

The circuit rank or cyclomatic number of is the minimum number of edges that have to be removed from to convert the graph into a tree.

Obviously, the circuit rank of is .

Lemma

The circuit rank of and are the same for .

Proof

From the definition of the subdivision graph and (1), the circuit rank of is .

Given the subdivision graph we label its nodes from to . Let be the degree of vertex , then denotes the diagonal degree matrix of and its adjacency matrix, defined as a matrix with the -entry equal to if vertices and are adjacent and otherwise.

We introduce now the probability transition matrix for random walks on or Markov matrix as . can be normalized to obtain a symmetric matrix .

(2)

The th entry of is .

Definition

The normalized Laplacian matrix of is

(3)

where is the identity matrix with the same order as .

We denote the spectrum of by . It is known that . The spectrum of the normalized Laplacian matrix of a graph provides us with relevant structural information about the graph, see Chung (1997); Butler (2016).

Below, we will then relate to some significant invariants of .

Definition

Replacing each edge of a simple connected graph by a unit resistor, we obtain an electrical network corresponding to . The resistance distance between vertices and of is equal to the effective resistance between the two equivalent vertices of  Klein and Randić (1993).

Definition

The multiplicative degree-Kirchhoff index of is defined as Chen and Zhang (2007):

(4)

This index is different from the classical Kirchhoff index Bonchev et al. (1994), , as it takes into account the degree distribution of the graph.

It has been proved Chen and Zhang (2007) that can be obtained from the spectrum of the normalized Laplacian matrix of :

(5)

where .

Thus, for , we have:

(6)

where are the eigenvalues of .

Definition

Given a graph , the Kemeny’s constant , also known as average hitting time, is the expected number of steps required for the transition from a starting vertex to a destination vertex, which is chosen randomly according to a stationary distribution of unbiased random walks on , see Hunter (2014) for more details.

It is known that is a constant as it is independent of the selection of starting vertex , see Levene and Loizou (2002). Moreover, the Kemeny’s constant can be computed from the normalized Laplacian spectrum in a very simple way as the sum of all reciprocal eigenvalues, except , see Butler (2016). Thus, we can write, for and :

(7)

The last graph invariant considered in this paper is the number of spanning trees of a graph . A spanning tree is a subgraph of that includes all the vertices of and is a tree. A known result from Chung Chung (1997) allows the calculation of this number from the normalized Laplacian spectrum and the degrees of all the vertices, thus the number of spanning trees of is

(8)

In the next section we provide an analytical expression for this invariant for any value of .

3 Normalized Laplacian spectrum of the subdivision graph

In this section we find an analytical expression for the spectrum of , the normalized Laplacian of the subdivision graph . We show that this spectrum can be obtained iteratively from the spectrum of . As , if is an eigenvalue of then is an eigenvalue of with the same multiplicity. We denote the multiplicity of as . Thus we calculate first the spectrum of .

Lemma

Let be any nonzero eigenvalue of and let . Then, is an eigenvalue of with the same multiplicity as .

Proof

Divide the vertices of into two groups and , where contains all the vertices created when the edges of are subdivided to generate and contains the rest. Obviously, has the same vertices as . Thus for convenience, in the following when any vertex of is considered, it also refers to the corresponding vertex of .

Let be any eigenvector associated to an eigenvalue of . Hence

(9)

Consider a vertex of and denote the set of all its neighbors in and the set of all its neighbors in . From the definition of subdivision graph, there exists a bijection between and . If we rewrite Eq. (9) as we have

(10)

For any vertex , we have a similar relation

(11)

where vertex is the other neighbor of vertex in . Combining Eq. (10) and Eq. (11) yields

(12)

Therefore,

(13)

Eq. (13) directly reflects that is an eigenvalue of , is one of its corresponding eigenvectors and can be totally determined by by using Eq. (11). Hence .

Suppose now that . Then there exists an extra eigenvector associated to without an associated eigenvector in . But Eq. (11) provides with its corresponding eigenvector in as is nonzero, in contradiction with our assumption. Thus and the proof is completed.


Lemma

Let be any eigenvalue of such that and let and . Then and are eigenvalues of . Besides, .

Proof

This lemma is a direct consequence of Lemma 3


Remark

Lemmas 3 and 3 indicate that any nonzero eigenvalues of can be obtained from the spectrum of . Due to the simple expression of , and , we easily find that each eigenvalue, except , of will generate two unique eigenvalues of . Thus eigenvalues of , and consequently of , are determined this way.

The Perron-Frobenius theorem Friedland et al. (2013) shows that the largest absolute value of the eigenvalues of is always 1. And because of the existence of a unique stationary distribution for random walks on , the multiplicity of the eigenvalue is always for any . Since , we also obtain for any . This can be further explained from the perspective of Markov chains. Since is a bipartite graph Asratian et al. (1998) containing no odd-length cycles for any , random walks on it are periodic with period , which means it takes an even number of steps to return to the starting vertex. Thus the smallest eigenvalue of the Markov matrix of is  Zhang (2004). But random walks on a general graph can be aperiodic Diaconis and Stroock (1991) if the graph has an odd-length cycle. Hence the multiplicity of the eigenvalue of depends on the structure of  Zhang (2004).

Lemmas 3 and 3 allow us to obtain the transition between the eigenvalues of the normalized Laplacian of at each iteration step. If is an eigenvalue of then is an eigenvalue of . Let , by Lemma 3, is an eigenvalue of if . This allows us to state the following lemma:

Lemma

Let be any eigenvalue of such that and let and . Then and are eigenvalues of and .


Definition

Let be any finite multiset of real number where for . The multisets and are defined as

(14)
(15)

Our main result in this section is the following theorem.

Lemma

The spectrum of , for , is:

(16)

When , . When , if G contains any odd-length cycle then , otherwise , where is the circuit rank of .

Proof

Combining Lemma 3, Lemma 3 and considering Remark 3, the multiplicity of the eigenvalue of can be determined indirectly:

(17)

Based on the previous results, it is obvious that if and only if and G contains an odd-length cycle, otherwise , which completes the proof.

This result allows us to state the main result of this section.

Theorem

The spectra of is obtained from of as:

(18)

for and where is the multiplicity of the eigenvalue of .

Due to the particularity of the eigenvalue of , we call it the exceptional eigenvalue Bajorin et al. (2008) of the family of matrices whose spectra show self-similar characteristics. For many other family of graphs Bajorin et al. (2008); Teplyaev (1998) with a similar self-similar property with respect to the spectra of their Markov matrices, the multiplicity of exceptional eigenvalues grows fast as increases. However, for the normalized Laplacian of subdivision graphs , the multiplicity of the only exceptional eigenvalue is always for .

4 Application of the spectrum of subdivision graph

In this section we use the spectra of , the normalized Laplacian of the subdivision graph , to compute some relevant invariants related to the structure of . Thus, we give closed formulas for the multiplicative degree-Kirchhoff index, Kemeny’s constant and the number of spanning trees of . These results depend only on and some invariants of the original graph .

4.1 Multiplicative degree-Kirchhoff index

Theorem

The multiplicative degree-Kirchhoff indices of and are related as follows, for any :

(19)

Therefore, the general expression for is

(20)

Proof

From Eq. (6) and Corollary 3, the relation between and can be expressed as:

(21)

provided that , where represents the eigenvalue of .

When and , we obtain:

(22)

The similarity of Eq. (21) and Eq. (22) indicates that Eq. (19) holds for any simple connected graph for .

Transforming Eq. (19) into

(23)

yields

(24)

We note here that this expression has been also obtained recently by Yang and Klein Yang and Klein (2015) by using a counting methodology not related with spectral techniques. Our result confirms both their calculation and the usefulness of the concise spectral methods described here.

4.2 Kemeny’s constant

Theorem

The Kemeny’s constant for random walks on can be obtained from through

(25)

The general expression is

(26)

Proof

Theorem 4.2 is an obvious consequence of Theorem 4.1 considering Eq. (7).

4.3 Spanning trees

Theorem

The number of spanning trees of is, for any :

(27)

Proof

From Eq. (8) and the properties of the subdivision of a graph:

(28)

Here are the eigenvalues of

Then, by the same techniques used in the previous subsection, we obtain

(29)

if .

When and , the relation becomes

(30)

which coincides with Eq. (29).

Therefore, the equality

(31)

holds for any as the circuit rank remains unchanged.

5 Conclusion

In this study we have focused on the analytical calculation of , the spectra for the normalized Laplacian of iterated subdivisions of any simple connected graph. This was possible through the analysis of the eigenvectors corresponding to adjacent vertices at different iteration steps. Our methods could be also applied to find the spectra of other graph families constructed iteratively.

The simple relationship between the spectrum of the Markov and the normalized Laplacian matrices of a subdivision graph facilitates the calculations to obtain the exact distribution and values of all eigenvalues. One particular interesting result is that the multiplicity of the exceptional eigenvalues of do not increase exponentially with as in Xie et al. (2015); Zhang et al. (2014) but is a constant determined by the value of the circuit rank of the initial graph.

The calculation of the multiplicative degree-Kirchhoff index, Kemeny’s constant and the number of spanning trees in section 4 is also succinct, if we compare it to other methods, thanks to the knowledge of the full spectrum of or . The expressions found can be directly used in the analysis of subdivisions of any simple connected graph while only needing minimal structural information from it, and this is part of the value of our study.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under grant No. 11275049. F.C. was supported by the Ministerio de Economia y Competitividad (MINECO), Spain, and the European Regional Development Fund under project MTM2014-60127.

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