Laplacian Distribution and Domination
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.
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
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  for connected and . This was recently improved in . We refer to the introduction of  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  shows that for trees , .
The following spectral lower bound for was proved in :
If is a graph, then .
In this paper we observe that for isolate-free one has
Since , this inequality generalizes the result in  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 , 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.
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 ) is always at most , we can find a graph for which . Similar examples can be found for the fractional domination number , and the irredundance number . 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 . In this paper, a star is the complete bipartite graph , and .
The star on vertices has Laplacian spectrum .
For graphs and where , and , we have .
Let denote the -th largest eigenvalue of a Hermitian matrix .
If and are Hermitian matrices of order , and is positive semi-definite, then , for .
Let and be graphs with . Then
for all , ;
for any , ;
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 ().
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 .
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,
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
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  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 . They proved that if is connected and , then .
Theorem 2 strengthens a recent result by Zhou, Zhou and Du  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
If is isolate-free then .
It seems interesting in its own right that
If is isolate-free, then .
When combined with a known lower bound on for trees, Theorem 1 implies something interesting about the interval .
If is a tree, then .
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  that for trees , is a lower bound for both and . For connected, it is also known  that , so Theorem 2 implies
For connected graphs , .
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 . A result of Jaeger and Payan  says that if is a graph then
and these bounds are tight. The following theorem by Cockayne and Hedetniemi characterizes when equality occurs in (1).
Theorem 3 ().
For any graph , with equality if and only if or .
We can use this to obtain the following:
For any graph , with equality if and only if or .
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
For any graph , .
We conclude that any graph of order must have fewer than Laplacian eigenvalues in at least one of the intervals or .
A Gallai-type theorem has the form where and are graph parameters. There are exactly Laplacian eigenvalues, so the equation
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  ).
For any graph , .
For any graph , .
if and only if .
Berge  gives an early bound for :
For any graph , .
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.
It is well-known that the decision problem DOMINATING SET is NP-complete , 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 . 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  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.
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  that for all graphs and ,
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, ).
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 .
If satisfies Vizing’s conjecture, then .
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]).
Let and be graphs with Laplacian spectra
respectively. Then the Laplacian spectrum of is
For any graphs and , .
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. ∎
For any graph and any , .
Let be the tree of order in Figure 1 for which
We claim that for all
There exists a sequence of graphs with .
Ratios for certain classes
Consider the two approximation ratios:
Let be a graph on vertices and edges, and let be the graph obtained by adding an edge. Then for any ,
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, . ∎
If is a tree, then .
A connected graph having edges is called c-cyclic. We can generalize Theorem 10 as follows.
If is -cyclic, , then .
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 , it is straightforward to determine that . Using the domination algorithm in  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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