Laplacian Distribution and Domination

Laplacian Distribution and Domination

Domingos M. Cardoso Departamento de Matemática, Univ. de Aveiro, 3810-193 Aveiro, Portugal dcardoso@ua.pt David P. Jacobs School of Computing, Clemson University Clemson, SC 29634 USA dpj@clemson.edu  and  Vilmar Trevisan Instituto de Matemática, UFRGS, 91509–900 Porto Alegre, RS, Brazil trevisan@mat.ufrgs.br
Abstract.

Let denote the number of Laplacian eigenvalues of a graph in an interval , and let denote its domination number. We extend the recent result , and show that isolate-free graphs also satisfy . In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of , showing that . However, for -cyclic graphs, . For trees , .

Key words and phrases: graph, Laplacian eigenvalue, domination number.

AMS subject classification: 05C50, 05C69.

Domingos M. Cardoso was partially supported by the Portuguese Foundation for Science and Technology (FCT–Fundação para a Ciência e a Tecnologia), through the CIDMA – Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.
David P. Jacobs and Vilmar Trevisan were supported by CNPq Grant 400122/2014-6, Brazil

1. Introduction

Let be an undirected graph with vertex set . For , its open neighborhood denotes the set of vertices adjacent to . The adjacency matrix of is the matrix for which if and are adjacent, and otherwise.

The Laplacian matrix of is defined as , where is the diagonal matrix in which , the degree of . The Laplacian spectrum of is the multi-set of eigenvalues of , we number

It is known that . Unless indicated otherwise, all eigenvalues in this paper are Laplacian. We refer to [22, 23] for more background on the Laplacian spectra of graphs.

A set is dominating if every is adjacent to some member in . The domination number is the minimum size of a dominating set. Its decision problem is well-known to be NP-complete, and it is even hard to approximate.

Since 1996, several papers have been written relating the Laplacian spectrum of a graph with . Often these results obtain a bound, involving , for a specific eigenvalue such as or . For example, it was shown that by Brand and Seifter [6] for connected and . This was recently improved in [26]. We refer to the introduction of [18] for a summary of these results.

Other spectral graph theory papers, including this one, are interested in distribution, that is, the number of Laplacian eigenvalues in an interval. For a real interval , denotes the number of Laplacian eigenvalues of in . There exist several papers in the literature that relate Laplacian distribution to specific graph parameters, including . For example, the paper by Zhou, Zhou and Du [27] shows that for trees , .

The following spectral lower bound for was proved in [18]:

Theorem 1.

If is a graph, then .

In this paper we observe that for isolate-free one has

Since , this inequality generalizes the result in [27] for trees.

Our paper seeks applications to the inequalities and . We also seek insight into the ratios of these numbers. In the examples given in [18], the numbers and were equal or differed by one. We will see that this does not happen in general.

The remainder of our paper is organized as follows. We finish this introduction by considering the sharpness of these inequalities. In the next section we recall the proof of Theorem 1 and modify it to obtain an inequality involving . In Section 3 we obtain several new results based on existing Nordhaus-Gaddum inequalities and Gallai-type theorems. One interesting new Nordhaus-Gaddum result is that for any graph , with equality if and only if or . Another interesting result is that a graph must have fewer than Laplacian eigenvalues in at least one of the intervals or . In Section 4, using results from the approximation literature, we explain why we can’t expect the quantities or to be close to . Using some results on Vizing’s conjecture, we show that . For trees, . For -cyclic graphs , , . These results seem interesting in light of the domination number’s general inapproximability. In Section 5 we observe that many results also hold for the signless Laplacian spectrum.

Tightness

We briefly discuss whether is the natural graph parameter bounded below by and above by . For example, one might ask if there exists a graph parameter for which

We considered three well-known graph parameters, each bounded above by , and observed that they are not always bounded below by . More precisely, while the 2-packing number (see [3]) is always at most , we can find a graph for which . Similar examples can be found for the fractional domination number [14], and the irredundance number [11]. We omit the details.

One can also ask if there exists a graph parameter for which

for isolate-free . Graph parameters for which include the independent domination number , the edge covering number , and the matching number . In the first two cases we can provide counter examples to show they are not necessarily bounded above by . Interestingly, we will see that , when is isolate-free.

2. Upper bound for

In this section we show how to modify the proof of Theorem 1 to obtain a new inequality. For convenience, we recall the facts used to prove Theorem 1. Proofs or references can be found in [18]. In this paper, a star is the complete bipartite graph , and .

Lemma 1.

The star on vertices has Laplacian spectrum .

Lemma 2.

For graphs and where , and , we have .

Let denote the -th largest eigenvalue of a Hermitian matrix .

Lemma 3.

If and are Hermitian matrices of order , and is positive semi-definite, then , for .

Lemma 4.

Let and be graphs with . Then

  1. for all , ;

  2. for any , ;

  3. for any , .

Let be a set of vertices, and . A vertex is an external private neighbor of (with respect to ) if . That is, is a neighbor of , but not a neighbor of any other member of .

Lemma 5 ([4]).

Any graph without isolated vertices has a minimum dominating set in which every member has an external private neighbor.

We will say that has a star forest , if there exists a sequence of pairwise vertex-disjoint subgraphs of , with , for all , . We emphasize that stars have order .

Lemma 6.

Any isolate-free graph with domination number has a star forest such that every belongs to exactly one star, and the centers of the stars form a minimum dominating set.

Theorem 1 is a spectral lower bound for . The key to its proof was to take the star forest that cover all vertices,

guaranteed by Lemma 6. By Lemma 1 , and so . By part (2) of Lemma 4 we have .

If instead of counting the smallest eigenvalue in each star we count the largest, we can also obtain a spectral upper bound for . Assume that is isolate-free. In the construction of , each star contains vertices. When , the star has eigenvalues . When , the star has eigenvalues . So for all . Since these are disjoint stars, . By Lemma 4, part (3), . We conclude that

Theorem 2.

If is an isolate-free graph, then .

We will use some ideas from our proof of Theorem 2 to establish Theorem 10 and Theorem 11, later in Section 4. However, there is actually an alternative and simpler proof to Theorem 2 which we sketch. Recall that the matching number , is the size of a largest set of independent edges in . We first claim that for any graph . To see this, let be the subgraph of consisting of disjoint ’s and isolated vertices. Then . By part (3) of Lemma 4, we must have . Finally, it is known [17] that if is isolate-free then , and so Theorem 2 follows.

A connection between and the number of Laplacian eigenvalues strictly greater than two was shown in 2001 by Ming and Wang [21]. They proved that if is connected and , then .

Theorem 2 strengthens a recent result by Zhou, Zhou and Du [27] which says that for trees , . Note that Theorem 2 requires be isolate-free while Theorem 1 does not. This happens because isolates in Theorem 1 can be disregarded as they increase both sides of the inequality by one. In Theorem 2 an isolate increases one side of the inequality but not the other. Theorem 1 and Theorem 2 imply

Corollary 1.

If is isolate-free then .

It seems interesting in its own right that

Corollary 2.

If is isolate-free, then .

When combined with a known lower bound on for trees, Theorem 1 implies something interesting about the interval .

Corollary 3.

If is a tree, then .

Proof.

We have

The first inequality follows by the bound for trees given in [5, Thr. 4.1]. The second inequality follows from Theorem 1. ∎

3. Applications

Recall that the distance between vertices and is the number of edges in a shortest path between them, and the graph’s diameter, , is the greatest distance between any two vertices. It is known [15] that for trees , is a lower bound for both and . For connected, it is also known [17] that , so Theorem 2 implies

Corollary 4.

For connected graphs , .

Nordhaus-Gaddum inequalities

A Nordhaus-Gaddum inequality is a bound on the sum or product of a parameter for a graph and its complement . For an overview of Nordhaus-Gaddum inequalities for domination-related parameters we refer to Chapter 10 in [17]. A result of Jaeger and Payan [19] says that if is a graph then

(1)
(2)

and these bounds are tight. The following theorem by Cockayne and Hedetniemi characterizes when equality occurs in (1).

Theorem 3 ([10]).

For any graph , with equality if and only if or .

We can use this to obtain the following:

Theorem 4.

For any graph , with equality if and only if or .

Proof.

From Theorem 1 and (1) we must have

(3)

for any . Since and , we must have equality if or . Conversely if , then (3) forces . By Theorem 3 it follows that or . ∎

From Theorem 1 and (2) we also have

Theorem 5.

For any graph , .

Recall [22, Theorem 3.6] that if has Laplacian eigenvalues

then the Laplacian eigenvalues of are:

It follows that . Then from Theorem 5

We have

Theorem 6.

For any graph , .

We conclude that any graph of order must have fewer than Laplacian eigenvalues in at least one of the intervals or .

Gallai-type theorems

A Gallai-type theorem has the form where and are graph parameters. There are exactly Laplacian eigenvalues, so the equation

(4)

can be regarded as a trivial Gallai-type theorem. A spanning forest of a graph is a spanning subgraph which contains no cycles. Let denote the maximum number of pendant edges in a spanning forest of .

Theorem 7 ( Nieminen [24] ).

For any graph , .

Corollary 5.

For any graph , .

Proof.

From Theorem 1 and (4) we know that

(5)

the left side being by Theorem 7. ∎

Corollary 6.

if and only if .

Proof.

This follows from (4) and Theorem 7. ∎

Berge [2] gives an early bound for :

(6)

where denotes the maximum vertex degree. In [12] the authors study when equality in (6) occurs. Combining (5) and (6) give

Theorem 8.

For any graph , .

As a simple application to Theorem 8, suppose we are given a list

of non-negative numbers and wish to know if there is a graph whose Laplacian spectrum is . Then Theorem 8 imposes a necessary condition on . Let . Any graph such that must have vertices whose degrees are bounded by .

4. Approximating

In this section we explain why it is hard to approximate with a polynomial computable spectral quantity of the form . We show that and do not even achieve logarithmic approximation ratios. Yet, for certain classes of graphs such as trees and -cyclic graphs, is bounded by a constant.

Inapproximability

It is well-known that the decision problem DOMINATING SET is NP-complete [13], even for planar graphs. In the approximation algorithm literature the problem is classified as class II in the taxonomy of NP-complete problems given in [1]. Roughly speaking, this means that approximating with better than a logarithmic ratio is hard. A problem is called quasi-NP-hard if a polynomial-time algorithm for it could be used to solve all NP problems in time . Thus the notion is slightly weaker than NP-hard.

Lund and Yannakakis [20, Thr. 3.6] showed that it is quasi-NP-hard to compute a polynomial-time function for which

when . Letting , we see this is equivalent to computing a polynomial time for which

Good approximations of do exist. The fractional domination number can be computed in polynomial time using linear programming. Given a vertex ordering, we can compute in polynomial time an approximation for using the greedy domination algorithm. Clearly for any graph ,

In [8] Chappell, Gimbel and Hartman proved that is in . It follows that both and must also be in . Note this result does not contradict that of Lund and Yannakakis, provided the constants of proportionality are sufficiently large.

Figure 1.

Example

We now construct an infinite sequence of graphs for which the ratio . Our construction uses the tree of order , shown in Figure 1. It is known [18] that and .

Recall that the Cartesian product of two graphs and is the graph with vertex set for which and are adjacent if and only if and or and .

In 1968 V. G. Vizing conjectured [25] that for all graphs and ,

(7)

While this currently remains an open problem, many partial results exist. We say that satisfies Vizing’s conjecture if (7) holds for all graphs . Many classes of graphs are known to satisfy Vizing’s conjecture.

Lemma 7 ( Theorem 8.2, [7]).

All trees satisfy Vizing’s conjecture.

It is easy to show that the Cartesian product is an associative operation. Let denote the Cartesian product of copies of .

Lemma 8.

If satisfies Vizing’s conjecture, then .

Proof.

By induction on , the case for being trivial. Assume that . Using the induction assumption, the fact that satisfies Vizing’s conjecture, and the associativity of , we have

completing the proof. ∎

The following is well-known (See, for example, [22, Thr. 3.5]).

Lemma 9.

Let and be graphs with Laplacian spectra

and

respectively. Then the Laplacian spectrum of is

Lemma 10.

For any graphs and , .

Proof.

By Lemma 9, Laplacian eigenvalues of are of the form , where and are eigenvalues of and respectively. A necessary condition for is that and . There are at most such pairs. ∎

Lemma 11.

For any graph and any , .

Proof.

The case is trivial, and is handled by Lemma 10. Assume . Then using Lemma 10 and the induction assumption, we have:

The right side is completing the induction. ∎

Let be the tree of order in Figure 1 for which

(8)

We claim that for all

(9)

The first inequality follows by Lemma 11, and the second inequality follows by Theorem 1. The third inequality follows by Lemma 7 and Lemma 8.

Theorem 9.

There exists a sequence of graphs with .

Proof.

We let . Using , (8) and (9) we have

Ratios for certain classes

Consider the two approximation ratios:

(10)
(11)

Both ratios can get arbitrarily large. By Theorem 9 the first of these ratios is not bounded by . The second ratio also gets arbitrarily large. When is the complete graph, we see that ratio (11) is .

Consider (11) for paths . It is well-known that . By Thr. 4.1 in [5] we also know , and so (11) is at most . Using ideas from Section 2, we show that for all trees ratio (11) is less than two.

Lemma 12.

Let be a graph on vertices and edges, and let be the graph obtained by adding an edge. Then for any ,

Proof.

Let and be the respective Laplacian spectra of and . By the well-know interlacing theorem [16, Thr. 2.4] for Laplacian eigenvalues we know

If , then . If then . We may assume that . Choose to be the largest index for which . Then . There is a single eigenvalue of , namely in . If , then . Otherwise, . ∎

Theorem 10.

If is a tree, then .

Proof.

Let be the star forest guaranteed by Lemma 6. Then is exactly . Starting with , we can construct by adding edges. By Lemma 12 the addition of each edge can increase by at most one. Therefore

But the right side is and the theorem follows. ∎

A connected graph having edges is called c-cyclic. We can generalize Theorem 10 as follows.

Theorem 11.

If is -cyclic, , then .

Proof.

Let be the star forest in from Lemma 6. Then we may select additional edges to form a spanning tree . Since has edges, there must be remaining edges. Therefore can be constructed from by adding edges. By Lemma 12

or

the last inequality holding because and . ∎

Let us now consider ratio (10) for trees. For the tree in Figure 1, the ratio (10) is . It is possible to generalize this example. We construct the tree on vertices by taking copies of this tree, and adjoining the root to each copy. Using the algorithm in [5], it is straightforward to determine that . Using the domination algorithm in [9] it can be shown that . Thus, the difference between grows arbitrarily large. However, the ratio (10) remains at . In all known examples of trees ratio (10) is either or , and it is tempting to conjecture that the ratio is bounded by a constant for trees.

5. Concluding remarks

Many of the results of this paper also apply to the signless Laplacian spectrum. For example, if we let denote the number of signless Laplacian eigenvalues of in , then Theorem 1 and Theorem 2 are also true if we replace with .

We conclude by suggesting two problems for further study. First, characterize those graphs for which . Second, determine if bounded by a constant for trees .

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