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Vol. 18 No. 2
Journal of Systems Science and Complexity
Apr., 2005
A RELATION BETWEEN THE MATCHING NUMBER
AND LAPLACIAN SPECTRUM OF A TREE∗
FAN Yizheng
(Department of Mathematics, Nanjing Normal University, Nanjing 210097, China;
?
School of Mathematics and Computational Science, Anhui University,
Hefei 230039, China. Email: [email protected])
Abstract. Let T be a tree with matching number µ(T ). In this paper we obtain the
following result: If T has no perfect matchings, then µ(T ) is a lower bound for the number
of nonzero Laplacian eigenvalues of T which are smaller than 2.
Key words. Tree, Laplacian spectrum, matching number.
1 Introduction
Let G = (V, E) be a simple graph with vertex set V = V (G) = {v1 , v2 , · · · , vn } and edge
set E = E(G). The adjacency matrix of G is denoted by A(G) = (aij ), where aij = 1 if vi and
vj are adjacent and aij = 0 otherwise. The degree diagonal matrix of G is denoted by D(G) =
diag(d1 (G), d2 (G), · · · , dn (G)), where di (G) is the degree of vi . Then L(G) = D(G) − A(G) is
the Laplacian matrix of G. It is known that L(G) is a singular, positive semidefinite symmetric
matrix. The eigenvalues of L(G) are called the Laplacian eigenvalues of G, and are denoted
by λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G) = 0. It is proved in [1, Theorem 1] that if λ is a Laplacian
eigenvalue of G, then 0 ≤ λ ≤ n, and the multiplicity of 0 equals the number of components of
G, the multiplicity of n equals one less than the number of components of the complement of
G. A subset M of E(G) is called a matching of G if any two edges in M are not incident. The
maximum of cardinalities of all matchings of G is commonly known as its matching number
denoted by µ(G). Clearly, n ≥ 2µ(G). If n = 2µ(G), we say G has perfect matchings. Let I
be an interval of the real line. The number of Laplacian eigenvalues of G, with multiplicities
counted, that belong to I, is denoted by mG (I). Especially, if I = {λ}, then mG (λ) is just the
multiplicity of λ as a Laplacian eigenvalue of G. R. Grone, R. Merris, V. S. Sunder[2] and R.
Merris[3,4] study the bounds of mG (I) for some certain I’s, especially for I = (2, n]. Recently,
G. J. Ming and T. S. Wang[5] give a lower bound for mG (2, n] in terms of the matching number
of G as follows.
Theorem 1.1[5] Let G be a graph on n vertices with n > 2µ(G). Then
mG (2, n] ≥ µ(G).
Received May 20, 2002.
Revised November 11, 2004.
*This research is supported by Anhui provincial Natural Science Foundation, Natural Science Foundation of
Department of Education of Anhui Province of China (2004kj027), the Project of Research for Young Teachers
of Universities of Anhui Province of China (2003jql01), and the Project of Anhui University for Talents Group
Construction.
? This is the current corresponding address of the author.
No. 2
RELATION BETWEEN MATCHING NUMBER AND SPECTRUM
Few results can be found for the bounds for mG (0, 2), except the following one in [2].
Theorem 1.2[2] If T is a tree on n vertices with diameter d, then
d
mT (0, 2) ≥
≤ mT (2, n],
2
175
(1)
where square brackets indicate the greatest integer function.
In this paper, we give a lower bound for mT (0, 2) for a tree T also in terms of its matching
number. In Section 2, we prove that if n > 2µ(T ), then mT (0, 2) ≥ µ(T ). Combining this
result and Theorem 1.1, if n > 2µ(T ), then
mT (0, 2) ≥ µ(T ) ≤ mT (2, n],
(2)
which has a symmetric form corresponding to result (1). Furthermore, the result (2) is a
generalization of result (1) to some extent. One may find from [5, Theorem 1] that if T is tree
on n ≥ 3 vertices, and T 6= Pi (i = 3, 4, 5), then mT (0, 2) ≥ 2, where Pi denotes a path on i
vertices. In Section 3, we determine all trees T with mT (0, 2) = 2.
2 A Lower Bound for mT (0, 2) for a Tree T
Let G be a graph and let G + e be the graph obtained from G by inserting a new edge e
into G. It follows from Courant-Weyl inequalities (see e.g. [6, Theorem 2.1]) that the following
is true.
Theorem 2.1 Let G be a graph on n vertices. Then the Laplacian eigenvalues of G interlace
those of G + e, that is,
λ1 (G + e) ≥ λ1 (G) ≥ λ2 (G + e) ≥ λ2 (G) ≥ · · · ≥ λn (G + e) = λn (G) = 0.
By Theorem 2.1, we get the following
Corollary 2.2 Let G be a connected graph on n vertices, and let G−e be the graph obtained
from G by removing one edge e from G. Let I = (0, r](or (0, r)), where r is a real number. If
G − e is disconnected, then
mG (I) ≥ mG−e (I).
Theorem 2.3[2] Suppose T is a tree on n vertices. If λ > 1 is an integer eigenvalue of
L(T ) with corresponding eigenvector u, then
1) λ | n (i.e. λ exactly divides n),
2) mT (λ) = 1,
3) no coordinate of u is zero.
Theorem 2.4[5] Let T be a tree on n vertices with n = 2µ(T ). Then
λµ(T ) (T ) = λ n2 (T ) = 2.
By Theorem 2.3 and Theorem 2.4, we get the following
Corollary 2.5 Let T be a tree on n vertices with n = 2µ(T ). Then
mT (0, 2) =
n
− 1 = mT (2, n].
2
176
Vol. 18
FAN YIZHENG
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Figure 1
To prove our main result, we first discuss two special classes H1 , H2 of trees in Figure 1,
where, in H1 , k ≥ 1, and for each i = 1, 2, · · · , k, Ti is a tree with perfect matchings; in H2 ,
m ≥ 1, and T1 is a tree with perfect matchings satisfying µ(T1 ) = µ(H2 ).
Lemma 2.6 For each graph H1 ∈ H1 and each graph H2 ∈ H2 of Figure 1, the following
hold:
mH1 (0, 2) = mH1 (2, +∞) = µ(H1 ),
mH2 (0, 2) ≥ µ(H2 ).
Proof We first prove the result for H1 ∈ H1 . If k = 1, by Theorem 2.1,
λµ(T1 )+1 (H1 ) ≤ λµ(T1 ) (H1 − e1 ) = λµ(T1 ) (T1 ).
By Theorem 2.4, λµ(T1 ) (T1 ) = 2. So λµ(T1 )+1 (H1 ) ≤ 2, and hence λµ(T1 )+1 (H1 ) < 2 by
Theorem 2.3 as H1 has odd number of veritces. By Theorem 1.1, λµ(T1 ) > 2, and the result
holds. Suppose the result on H1 holds for k = l − 1(l ≥ 2). Then, for k = l,
mH1 −T1 (0, 2) = mH1 −T1 (2, +∞) = µ(T2 ) + · · · + µ(Tl ),
where H1 − T1 is the graph obtained from H1 by removing all vertices of T1 together with all
edges incident to any of them. Hence by Theorem 2.4 and Corollary 2.5,
mH1 −e1 (0, 2) = mH1 −e1 (2, +∞) =
l
X
µ(Ti ) − 1 = µ(H1 ) − 1;
λµ(H1 ) (H1 − e1 ) = 2.
i=1
By a similar discussion, we have λµ(H1 )+1 < 2 and λµ(H1 ) > 2. The result holds by induction.
For the graph H2 ∈ H2 , the result holds for m = 1 by the above discussion. Otherwise, by
Corollary 2.2, the result also holds by a sequential removal of edges e2 , e3 , · · · , em .
The following is our main result. Note that a vertex of G is called pendant if it is of degree
1; and a vertex of G is called quasipendant if it is adjacent to a pendant vertex.
Theorem 2.7 Let T be a tree on n vertices with n > 2µ(T ). Then
mT (0, 2) ≥ µ(T ).
No. 2
RELATION BETWEEN MATCHING NUMBER AND SPECTRUM
177
Proof Let M (T ) be a matching of T with µ(T ) edges, and let H be the subgraph of
T induced by all the vertices incident to edges in M (T ). Then the subgraph induced by
V (T )\V (H) consists of isolated vertices. Let the components of H be T1 , T2 , · · · , Tk (k ≥ 1),
and let the vertices of V (T )\V (H) be v1 , v2 , · · · , vm (m ≥ 1). Then T1 , T2 , · · · , Tk are all trees
with perfect matchings.
We now construct a new tree T 1 on k + m vertices from T − H by inserting new vertices
u1 , u2 , · · · , uk and new edges e such that e = (ui , vj ) ∈ T 1 if and only if there is an edge in T
joining some vertex of Ti and the vertex vj (i.e. T 1 is obtained from T by contracting each tree
Ti into a vertex ui ). If T 1 is a star(a tree with k + m − 1 pendant vertices), then T belongs
to one class of Figure 1. By Lemma 2.6, the result holds. Otherwise, by a sequential removal
of edges of T 1 joining a quasipendant vertex and a non-pendant vertex, we get a graph whose
components are stars or not. If some component of the obtained graph is not star, repeat the
above procedure. At last, we get a graph T 2 whose components F1 , F2 , · · · , Fp (p ≥ 2) are all
stars by removal of some edges of T 1 which do not belong to H. So a new graph Tb can be
obtained from T by removing some edges of T which do not belong to H, and the components
of Tb are E1 , E2 , · · · , Ep belonging to class of H1 or H2 in Figure 1. It is easily seen that
µ(T ) = µ(Tb) = µ(E1 ) + µ(E2 ) + · · · + µ(Ep ). By Lemma 2.6, mEi (0, 2) ≥ µ(Ei ), i = 1, 2, · · · , p.
Then, by Corollary 2.2,
mT (0, 2) ≥ mTb(0, 2) = mE1 (0, 2) + mE2 (0, 2) + · · · + mEp (0, 2) ≥ µ(T ),
and the result follows.
By Corollary 2.5, Theorem 1.1 and Theorem 2.7, we get the following result, which has a
symmetric form corresponding to result (1) of Theorem 1.2.
Corollary 2.8 Let T be a tree on n vertices. Then
mT (0, 2) = µ(T ) − 1 = mT (2, n], if
n = 2µ(T ),
mT (0, 2) ≥ µ(T ) ≤ mT (2, n],
n > 2µ(T ).
if
Remark Let T be a tree on n vertices with diameter d. Then µ(T ) ≥ [d/2] and µ(T ) ≥
[d/2] + 1 if n = 2µ(T ). So Corollary 2.8 can be considered as a generalization of Theorem 1.2.
3 Trees T with mT (0, 2) = 1 or 2
By [5, Theorem 1] and Corollary 2.8, the following is easily obtained.
Theorem 3.1 Let T be a tree on n ≥ 3 vertices. Then mT (0, 2) = 1 if and only if T is P3
or P4 .
Theorem 3.2 Let T be a tree on n ≥ 3 vertices. Then mT (0, 2) = 2 if and only if T is one
of the following graphs: a star on 4 vertices, P5 , P6 , and S1 , S2 in Figure 2.
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Figure 2
Proof By Corollary 2.8, the necessity holds. If mT (0, 2) = 2, then µ(T ) ≤ 3 by Corollary
2.8. If µ(T ) = 1, then T is a star. Thus the nonzero Laplacian eigenvalues of T are 1 with
178
FAN YIZHENG
Vol. 18
multiplicity n − 2 and n with multiplicity 1. So T is a star on 4 vertices. If µ(T ) = 3, then
n = 2µ(T ) = 6. So T is P6 or S2 in Figure 2. For the last case of µ(T ) = 2, T has a
path P4 as its subgraph since T cannot be a star. So the subgraph induced by V (T )\V (P4 )
consists of isolated vertices. Let p(T ) be the number of pendant vertices of T . It is easily seen
that mT [0, 1] ≥ p(T ) by Cauchy interlacing inequalities. So p(T ) ≤ 3, and consequently the
cardinality of V (T )\V (P4 ) is at most 3. Thus 5 ≤ n ≤ 7. If n = 7, then µ(T ) = 3 as p(T ) ≤ 3,
which contradicts our assumption. If n = 5, then T is P5 or S1 in Figure 2. If n = 6, then T
is a graph obtained from P5 by inserting a new vertex and an edge joining the inserted vertex
and one quasipendant vertex of the path. By a direct calculation, this is impossible. The result
holds.
References
[1] W. N. Anderson and T. D. Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear
Algebra, 1985, 18: 141–145.
[2] R. Grone, R. Merris and V. S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal.
Appl., 1990, 11: 218–238.
[3] R. Merris, The number of eigenvalues greater than two in the Laplacian spectrum of a graph,
Portugal. Math., 1991, 148: 345–349.
[4] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl., 1994, 197/198: 143–176.
[5] G. J. Ming and T. S. Wang, A relation between the matching number and the Laplacian spectrum
of a graph, Linear Algebra Appl., 2001, 325: 71–74.
[6] D. M. Cvetković, M. Doob and H. Sachs, Spectra of Graphs-Theory and Applications(2nd ed.),
VEB Deutscher Verlag d.Wiss., Berlin, 1982.