Two Laplacians for the distance matrix of a graph

Linear Algebra and its Applications 439 (2013) 21–33

Contents lists available at SciVerse ScienceDirect

Linear Algebra and its Applications journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / l a a

Two Laplacians for the distance matrix of a graph Mustapha Aouchiche ∗ , Pierre Hansen GERAD and HEC Montreal 3000, chemin de la Côte-Sainte-Catherine Montréal (Québec), Canada H3T 2A7

ARTICLE INFO

ABSTRACT

Article history: Received 12 April 2012 Accepted 22 February 2013 Available online 30 March 2013

We introduce a Laplacian and a signless Laplacian for the distance matrix of a connected graph, called the distance Laplacian and distance signless Laplacian, respectively. We show the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. There is also an equivalence between the Laplacian spectrum and the distance Laplacian spectrum of any connected graph of diameter 2. Similarities between n, as a distance Laplacian eigenvalue, and the algebraic connectivity are established. © 2013 Elsevier Inc. All rights reserved.

Submitted by R.A. Brualdi AMS classification: Primary 05C50 Keywords: Distance matrix Eigenvalues Laplacian Signless Laplacian Graph

1. Introduction We begin by recalling some definitions. In this paper, we consider only simple, undirected and finite graphs, i.e, undirected graphs on a finite number of vertices without multiple edges or loops. A graph is (usually) denoted by G = G(V , E ), where V is its vertex set and E its edge set. The order of G is the number n = |V | of its vertices and its size is the number m = |E | of its edges. The adjacency matrix A of G is a 0–1 n × n-matrix indexed by the vertices of G and defined by aij = 1 if and only if ij ∈ E. Denote by (λ1 , λ2 , . . . , λn ) the A-spectrum of G, i.e., the spectrum of the adjacency matrix of G, and assume that the eigenvalues are labeled such that λ1 ≥ λ2 ≥ · · · ≥ λn . The matrix L = Diag (Deg ) − A, where Diag (Deg ) is the diagonal matrix whose diagonal entries are the degrees in G, is called the Laplacian of G. Denote by (μ1 , μ2 , . . . , μn ) the L-spectrum of G, i.e., the spectrum of the Laplacian of G, and assume that the eigenvalues are labeled such that μ1 ≥ μ2 ≥ · · · ≥ μn = 0. The matrix Q = Diag (Deg ) + A is called the signless Laplacian of G. Denote by (q1 , q2 , . . . , qn ) the Q -spectrum of ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Aouchiche), [email protected] (P. Hansen). 0024-3795/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.02.030

22

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

Fig. 1. The transmission regular but not degree regular graph with the smallest order.

G, i.e., the spectrum of the signless Laplacian of G, and assume that the eigenvalues are labeled such that q1 ≥ q2 ≥ · · · ≥ qn . Given two vertices u and v in a connected graph G, d(u, v) = dG (u, v) denotes the distance (the length of a shortest path) between u and v. The Wiener index W (G) of a connected graph G is defined to be the sum of all distances in G, i.e., 1  d(u, v). W (G) = 2 u,v∈V The transmission Tr (v) of a vertex v is defined to be the sum of the distances from v to all other vertices in G, i.e.,  d(u, v). Tr (v) = u∈V

A connected graph G = (V , E ) is said to be k-transmission regular if Tr (v) = k for every vertex v ∈ V . The transmission regular graphs are exactly the distance-balanced graphs introduced in [29]. They are also called self-median graphs [9]. It is clear that any vertex-transitive graph (a graph G in which for every two vertices u and v, there exist an automorphism f on G such that f (u) = f (v)) is a transmission regular graph. Indeed, the graph on 9 vertices illustrated in Fig. 1 is 14-transmission regular graph but not degree regular and therefore not vertex-transitive. For more examples of transmission regular but not degree regular graphs see [1,2]. As usual, we denote by Pn the path, by Cn the cycle, by Sn the star, by Ka,n−a the complete bipartite graph and by Kn the complete graph, each on n vertices. The graph obtained from a star Sn , n ≥ 3, by adding and edge is well defined and here denoted by Sn+ . The distance matrix D of a connected graph G is the matrix indexed by the vertices of G where Di,j = d(vi , vj ) and d(vi , vj ) denotes the distance between the vertices vi and vj . Let ∂1 ≥ ∂2 ≥ · · · ≥ ∂n denote the spectrum of D. It is called the distance spectrum of the graph G. Among the first results related to the distance matrix, the remarkable theorem proved by Graham and Pollack [26] that gives a formulas of the determinant of the distance matrix of a tree depending only on the order n. The determinant is given by Det (D) = (−1)n−1 (n − 1)2n−2 . An immediate consequence of the above result is that the distance spectrum of any nontrivial tree on n vertices contains one positive eigenvalue and n − 1 negative eigenvalues. The impressive and promising result of Graham and Pollack [26] attracted much interest among algebraic graph theory researchers. Graham and Lovász [25] computed the inverse of the distance matrix of a tree (see Theorem 3.2 bellow). Edelberg et al. [18], Graham and Lovász [25], and Hosoya et al. [30] studied the characteristic polynomial. Actually, they calculated certain, and in some cases all, the coefficients of the distance characteristic polynomial. Mihalic´ et al. [36] gave an overview of the use of the distance matrix in chemical graph theory. For more details about the distance spectrum see the Merris survey [35] as well as the references therein. Almost all results obtained for the distance

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

23

Fig. 2. The Petersen graph and its different spectra.

matrix of trees were extended to the case of weighted trees by Bapat [4] and Bapat et al. [5]. Extensions were done not only concerning the class of graphs but also regarding the distance matrix itself. Indeed, Bapat et al. [6] generalized the concept of the distance matrix to that of q-analogue of the distance matrix Dq = (dij ) defined, for a real number q, by ⎧ ⎨ 1 + q + q2 + · · · + qk−1 if d(i, j) = k > 0; dij = ⎩0 if i = j. We speak about a generalization of the distance matrix since D = D1 . Properties of Dq and its similarities with D are studied in [6]. In the present paper, we investigate another generalization of the distance matrix. We use an approach similar to that used for the adjacency matrix by defining a Laplacian and a signless Laplacian. Similarly to the (adjacency) Laplacian, we define the distance Laplacian of a connected graph G as the matrix DL = Diag (Tr )− D, where Diag (Tr ) denotes the diagonal matrix of the vertex transmissions in G. Let ∂1L ≥ ∂2L ≥ · · · ≥ ∂nL denote the spectrum of DL . We call it the distance Laplacian spectrum of the graph G. Along these lines, we define the distance signless Laplacian of a connected graph G to Q Q be DQ = Diag (Tr ) + D. Let ∂1 ≥ ∂2 ≥ · · · ≥ ∂nQ denote the spectrum of DQ . We call it the distance signless Laplacian spectrum of the graph G. In Fig. 2, we give a graph with its different spectra. G (t ), PLG (t ) and PQG (t ) denote the distance, the distance Laplacian and For a connected graph G, let PD the distance signless Laplacian characteristic polynomials respectively. With the help of the software Maple and using different properties of determinants and eigenvalues, we established the characterG (t ), PLG (t ) and PQG (t ) for some particular graphs. We next list them. istic polynomials PD 1.1. The complete graph The distance, the distance Laplacian and the distance signless Laplacian spectra of the complete graph Kn are respectively its adjacency, Laplacian and signless Laplacian spectra, i.e., K

PDn (t ) = (t K

− n + 1)(t + 1)n−1 ;

PL n (t ) = t (t K

PQn (t ) = (t

− n)n−1 ;

− 2n + 2)(t − n + 2)n−1 .

1.2. The complement of an edge The distance, the distance Laplacian and the distance signless Laplacian spectra of the complement of an edge Kn − e are respectively   ⎛ ⎞⎛ ⎞ n − 1 + (n − 1)2 + 8 n − 1 − (n − 1)2 + 8 Kn −e ⎠⎝t − ⎠(t + 2)(t + 1)n−3 ; PD (t ) = ⎝t − 2 2 K

PL n

−e

(t ) = t (t − n − 2)(t − n)n−2 ;

(1)

24

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33 K −e PQn (t )



= t−

3n − 2 +

√

n2

− 4n + 20



2

t

−

3n − 2 −

√

n2

− 4n + 20

2



(t − n + 2)n−2 .

1.3. The complete bipartite graph The distance, distance Laplacian and the distance signless Laplacian characteristic polynomials of the complete bipartite graph Ka,b are respectively

    Ka,b PD (t ) = t − n + 2 − a2 − ab + b2 t − n + 2 + a2 − ab + b2 (t + 2)n−2 ; Ka,b

PL

(t ) = t (t − n)(t − (2n − a))b−1 (t − (2n − b))a−1 ;

Ka,b PQ (t )

⎛

= ⎝t −

5n − 8 +

 9(a − b)2 2

+ 4ab

⎞⎛ ⎠ ⎝t

−

5n − 8 −

 9(a − b)2

+ 4ab

2

⎞ ⎠ (2)

× (t − 2n + b + 4)a−1 (t − 2n + a + 4)b−1 . 1.4. The graph Sn+

The distance, distance Laplacian and the distance signless Laplacian characteristic polynomials of the graph Sn+ are respectively  S+ PDn (t ) = t 3



− (2n − 7)t 2 − (7n − 17)t − (3n − 5) (t + 1)(t + 2)n−4 ;

S+

PL n (t ) = t (t − n)(t − 2n + 3)(t − 2n + 1)n−3 ;    S+ PQn (t ) = t 3 − (7n − 15)t 2 + 14n2 − 36n + 72 t





− 8n3 − 52n2 + 108n − 68

×(t − 2n + 5)n−3 . The rest of the paper is organized as follows. In Section 2, we present a brief overview of the graph cospectrality with respect to adjacency, Laplacian, signless Laplacian, normalized Laplacian and distance matrices. We also summarize the results of our enumeration of the trees on at most 20 vertices in order to test cospectrality with respect to D, DL and DQ . In Section 3, we discuss similarities between the distance, the distance Laplacian, the distance signless Laplacian, and the Laplacian spectra of a connected graph. We prove that the distance, the distance Laplacian, and the distance signless Laplacian spectra are equivalent on the class of transmission regular graphs with a fixed order. The equivalence between the Laplacian and the distance Laplacian spectra on the set of connected graphs with diameter 2 is also proved. Thereafter, we show that the interlacing theorem does not apply for the distance (signless) Laplacian spectrum. We prove that the distance signless Laplacian eigenvalues do not decrease by the deletion of an edge. In Section 4, we show that the second smallest distance Laplacian eigenvalue ∂nL−1 of G is for G exactly what the second smallest Laplacian eigenvalue of G is for G, i.e., ∂nL−1 of G can be seen as the algebraic connectivity of G. 2. Cospectrality Graphs with the same spectrum with respect to an associated matrix M are called cospectral graphs with respect to M, or M-cospectral graphs. Two M-cospectral non-isomorphic graphs G and H are called M-cospectral mates or M-mates. The question “which graph is defined with its A-spectrum” raised by Günthard and Primas [24] in 1956 in a paper relating spectral theory of graphs and Hückel’s theory from chemistry. It was conjectured [24] that there are no A-cospectral mates. A year later, the conjecture was refuted by Collatz and Sinogowitz [11] giving two A-cospectral mates, which are in fact, the smallest

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

25

Fig. 3. (a) Two A-cospectral trees. (b) Two A-cospectral graphs. (c) Two A-cospectral connected graphs.

Fig. 4. (a) Two L-cospectral graphs. (b) Two Q -cospectral graphs. (c) Two L-cospectral connected graphs.

such trees (see Fig. 3(a)). For the class of general graphs, the A-cospectral mates with the smallest order, first given by Cvetkovic´ [12], are illustrated in Fig. 3(b). The A-cospectral mates with smallest order among connected graphs, first given by [3], are illustrated in Fig. 3(c). Several constructions of A-cospectral mates were proposed in the literature. The first infinite family of pairs of A-cospectral mates, among trees, was constructed by Schwenk [38]. For more construction methods see for example [22,23,27,31,40,41]. The number of graphs with a mate also attracted much attention. Schwenk [38] proved that asymptotically every tree has a mate, i.e., the proportion of trees (among the class of trees) with mates tend to 1 when the order n tend to +∞. Such a statement is neither proved nor refuted for the class of graphs in general. Till now, computational experiments were done on the set of all graphs on up to 12 vertices [7]. Here, we partially reproduce the table, from [7], containing the number of graphs with a mate. Recall that there are no A-cospectral graphs with less than 5 vertices. Number of vertices

5

6

7

8

Number of graphs

34

156

1044

12346

Number of graphs with a mate

2

10

110

1722

Number of vertices

9

10

11

12

Number of graphs

274668

12005168

1018997864

165091172592

Number of graphs with a mate

51039

2560606

215331676

31067572481

Cospectrality with respect to other graph matrices were also the subject of many publications. The L-cospectrality is studied in [21,27,28,33,34,39]. The smallest L-cospectral graphs, with respect to the order, contain 6 vertices, and are given in Fig. 4(a). The Q -cospectrality is studied in [27,40] (see also [15]). The Q -cospectral graphs with smallest order are illustrated in Fig. 4(b). Another Laplacian matrix of a graph is defined by L = (Diag (Deg ))− 2 L(Diag (Deg ))− 2 (see [10] for more information). The L-cospectrality is studied in [8], and the smallest cospectral graphs, with respect to the order, are shown in Fig. 4(c). We said above that Schwenk [38] proved that the proportion of trees with A-mate is asymptotically 1. A similar result about D-mates is established by McKay [32]. The smallest (see Fig. 5) D-cospectral trees contain 17 vertices, and belong to an infinite family of pairs of D-mates that can be constructed using McKay’s method described in [32]. In fact, these two trees are the only D-cospectral trees on 17 vertices. Using Nauty (a computer program for generating graphs available at http://cs.anu.edu.au/ ∼bdm/nauty/), we generated all trees with at most 20 vertices and test cospectrality with respect 1

1

26

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

Fig. 5. The smallest D-cospectral trees.

Fig. 6. Two pairs of D-cospectral trees on 19 vertices.

Fig. 7. Five pairs, by column, of D-cospectral trees on 20 vertices.

to D, DL and DQ . Among the 123867 trees on 18 vertices, there are two pairs of D-mates. Note that these pairs can be obtained using McKay’s method. Among the 317955 trees on 19 vertices, there are six pairs of D-mates four of which can be obtained using McKay’s method. The remaining two pairs are given in Fig. 6. There are 14 pairs of D-mates over all the 823065 trees on 20 vertices. Nine pairs can be obtained using McKay’s method and the 5 remaining ones are presented in Fig. 7. Now, returning to our main subject, are there any cospectral trees with respect to the distance Laplacian or distance signless Laplacian? After the enumeration of all 1346023 trees on at most 20 vertices, we found no DL -mates and no DQ -mates. Thus a question naturally arises: can every tree be characterized by its DL -spectrum and/or DQ -spectrum. Cospectrality plays an important role in isomorphism theory. Actually, it is not yet known if testing isomorphism of two graphs is a hard problem or not, while determining whether two graphs are cospectral can be done in a polynomial time. Thus checking isomorphism is done among cospectral graphs only. The important motivation for introducing the study of the distance Laplacian and the distance signless Laplacian spectra of graphs comes from cospectrality considerations, and therefore graph isomorphism theory. The results of our experiments (limited to the class of trees on at most 20

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

27

vertices) led us to conjecture that every tree is determined by its distance Laplacian spectrum, and by its distance signless Laplacian spectrum, or at least to ask formally the question: can every tree determined by its DL -spectrum and/or by its DQ -spectrum? Naturally, the question can also be asked for connected graphs. 3. Similarities with other spectra In [15–17], Cvetkovic´ and Simic´ studied the spectral graph theory based on the signless Laplacian matrix. Among other results, they showed equivalence between the spectrum of the signless Laplacian and

• the adjacency spectrum for the class of (degree) regular graphs; • the Laplacian spectrum for the class of (degree) regular graphs; • the Laplacian spectrum for the class of bipartite graphs. Along these lines, we studied similarities between the spectra of different distance matrices associated to connected graphs. A first result is that there is equivalence between the spectra of the distance matrix D, the distance Laplacian DL , and the distance signless Laplacian DQ on the set of transmission regular graphs. Indeed, for a k-transmission regular graph G on n vertices, the distance Laplacian characteristic polynomial is given by PL (t )

= det (L − tI ) = (−1)n det (D − (k − t )I ) = (−1)n PD (k − t ),

and distance signless Laplacian characteristic polynomial of G is given by PQ (t ) = det (DQ

− tI ) = det (Diag (Tr ) + D − tI ) = det (kI + D − tI ) = det (D − (t − k)I )

= PD (t − k). Thus, if ∂1

≥ ∂2 ≥ · · · ≥ ∂n the distance spectrum of G, then

• k − ∂n ≥ k − ∂n−1 ≥ · · · ≥ k − ∂1 is the distance Laplacian spectrum of G; • k + ∂1 ≥ k + ∂2 ≥ · · · ≥ k + ∂n is the distance signless Laplacian spectrum of G. Using the above observation, one can calculate the distance Laplacian and the distance signless Laplacian characteristic polynomials of a transmission regular graph from its distance characteristic polynomial. For instance, we can do so for the cycle on n vertices. The distance characteristic polynomial of Cn was given in [20], according to the parity of n, as follows. If n = 2p (i.e., even) p     n2 π(2j − 1) C t + csc2 PDn (t ) = t p−1 · t − · . 4 n j=1 If n

= 2p + 1 (i.e., odd) 

C

PDn (t )

= t−

n2

−1



4

·

p  j=1

t

+

1 4

sec2

Since Cn is a k-transmission regular graph with k we derive the following:

πj n

  p  t · j=1

+

1 4

 csc2

π(2j − 1)



2n

= n2 /4 if n is even and k = (n2 − 1)/4 if n is odd,

• The distance signless Laplacian characteristic polynomial of the cycle Cn is as follows. If n = 2p (i.e., even)  C PL n (t )

=t· t−

n2 4

p−1

·

p  j=1

 t

−

n2 4

.



− csc

2

π(2j − 1) n



;

28

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

if n

= 2p + 1 (i.e., odd) C PL n (t )

= −t · ·

p 

p 

 t

j=1



t

j=1

−

− 2

n

n2

−1

−

4

−1 4

−

1 4

1 4

sec 

csc

2

2

πj



n

π(2j − 1)



.

2n

• The distance signless Laplacian characteristic polynomial of the cycle Cn is as follows. If n = 2k (i.e., even)  C PQn (t )

If n

n2

= t−

k−1 

n2

· t−

4



2

·

k 



j=1

t

−

n2 4



+ csc

2

π(2j − 1) n



.

= 2k + 1 (i.e., odd)  C PQn (t )

= t− ·

k 

n2

−1

·

2



j=1



t

−

n2

k 

t

j=1

−1 4



+

1 4

−

n2

csc2

−1

4 

+

1 4

sec

π(2j − 1) 2n

2

πj



n



.

The second equivalence established is between L-theory and DL -theory on the set of graphs of diameter 2.

≤ 2. Let λL1 ≥ λL2 ≥ · · · λLn−1 > = 0 be the Laplacian spectrum of G. Then the distance Laplacian spectrum of G is 2n − λLn−1 ≥ 2n − λLn−2 ≥ · · · 2n − λL1 > ∂nL = 0. Moreover, for every i ∈ {1, 2, . . . , n − 1} the eigenspaces corresponding to λLi and to 2n − λLn−i are the same. Theorem 3.1. Let G be a connected graph on n vertices with diameter D

λLn

Proof. Concerning the zero eigenvalue and in the case of D = 1 (corresponding to the complete graph), the result is trivial. For a connected graph of diameter 2, the transmission of each vertex v ∈ V is Tr (v)

= d(v) + 2(n − 1 − d(v)) = 2n − d(v) − 2,

where d(v) denotes the degree of v in G. Also, the distance matrix is D

= 2J − 2I − A,

where J is the all ones matrix. Thus the distance Laplacian matrix can be written as DL

= Diag (Tr ) − D = (2n − 2)I − Diag (Deg ) − 2J + 2I + A = 2nI − 2J − L.

Consider any eigenvalue λLi of the Laplacian L with 1 ≤ i ≤ n − 1, i.e., a nonzero Laplacian eigenvalue of G, and let Ui denote a Laplacian eigenvector for λLi . Since L is symmetric and the all ones column vector e is an eigenvector for λLn = 0, we have eT · Ui = 0 and therefore, J · Ui = 0. Thus D L · Ui

= 2nUi − L · Ui = (2n − λLi )Ui ,

which means that 2n − λLi is an eigenvalue of DL and Ui is a corresponding eigenvector. 

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

29

Before the stating an immediate corollary of the above theorem, we recall a result from graph theory literature that makes connection between the distance matrix D and the (adjacency) Laplacian L of a tree. Actually, Graham and Lovász [25] computed the inverse of the distance matrix of a tree in terms of the degrees of the vertices. The theorem is as follows. Theorem 3.2 [25]. If T is a tree on n ≥ 2 vertices with distance matrix D     (−1) A = aij , then the inverse matrix of D, D−1 = dij is given by ⎧

(−1)

dij

(2 − di )(2 − dj ) ⎨ − d2i + aij = ⎩ 2(n − 1) 2

if i

=j

if i

 = j,





= dij and adjacency matrix

where di denotes the degree of the vertex vi . Let X = [x1 , x2 , . . . , xn ] be the vector defined by xi = 2 − di for i = 1, 2, . . . , n, where di denotes the degree of the vertex vi . Using this notation, the formula of the inverse of D given in Theorem 3.2 can be written as D−1

=

1 2(n − 1)

X · Xt

1

− L. 2

(3)

The formulation (3) illustrates well the connection between the (inverse of) distance matrix of a tree and its Laplacian. Note that Theorem 3.2 was extended by Bapat, Kirkland and Neumann [5] to the case of weighted trees. They used the same formulation as in (3). Using Theorem 3.1, we can derive a relation between the distance Laplacian matrix of a graph of diameter 2 and its (adjacency) Laplacian. Actually, we can get any of the two matrices from the other using rotations defined by the eigenvectors of both matrices. Corollary 3.3. Let G be a graph on n Then DL

≥ 2 vertices with diameter 2, Laplacian L and distance Laplacian DL .

= 2nI − PRt LRP t ,

where P and R are orthogonal matrices defined by P = [P1 , P2 , . . . , Pn−1 , Pn ] and R = [Pn−1 , Pn−2 , . . . , P1 , Pn ] with P1 , P2 , . . . , Pn are the eigenvectors of DL corresponding to ∂1L , ∂2L , . . . , ∂nL respectively. Proof. From the proof of Theorem 3.1, the eigenvectors P1 , P2 , . . . , Pn belong to the eigenvalues ∂1L , ∂2L , . . . , ∂nL , respectively and with respect to the matrix DL , and to the eigenvalues λLn−1 , λLn−2 , . . . , λ1 , λn respectively and with respect to the matrix L. Then, we have DL

= PDiag (∂1L , ∂2L , . . . , ∂nL )P t = P (2nI − Diag (λLn−1 , λLn−2 , . . . , λ1 , λn )P t = 2nI − PRt LRP t .



It is well-known that both Laplacian and signless Laplacian eigenvalues of graph interlace with respect to an edge deletion in the following way. Let G be a graph and e an edge of G. Consider the graph G = G − e obtained from G by the deletion of the edge e. Let μ1 ≥ μ2 ≥ · · · ≥ μn and q1 ≥ q2 ≥ · · · ≥ qn (resp. μ 1 ≥ μ 2 ≥ · · · ≥ μ n and q 1 ≥ q 2 ≥ · · · ≥ q n ) be the Laplacian and signless Laplacian eigenvalues of G (resp. G ), respectively. Then μ1 ≥ μ 1 ≥ μ2 ≥ μ 2 ≥ · · · ≥ μn ≥ μ n (see [37]) and q1 ≥ q 1 ≥ q2 ≥ q 2 ≥ · · · ≥ qn ≥ q n (see [14]). Such interlacing does not apply to the distance Laplacian spectrum of a graph as it does to the Laplacian and signless Laplacian spectra. Indeed, consider the path Pn obtained from the cycle Cn by the deletion of an edge. The distance Laplacian spectra of Pn and Cn do not interlace for n ≥ 5. For instance the distance Laplacian spectrum of P6 is approximately (21.3929, 15, 12.8532, 11, 9.7539, 0) while the distance Laplacian spectrum of C6 is (13, 13, 10, 9, 9, 0).

30

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

As well as in the case of the distance Laplacian, the eigenvalues interlacing does not apply to the distance signless Laplacian. Indeed, consider the path Pn obtained from the cycle Cn by the deletion of an edge. The distance signless Laplacian spectra of Pn and Cn do not interlace for n ≥ 5. For instance the distance signless Laplacian spectrum of P6 is approximately (25.0838, 12.1755, 11.1743, 8.6727, 7.7418, 5.5118) while the distance signless Laplacian spectrum of C6 is (18, 9, 9, 8, 5, 5). The corresponding property for the distance Laplacian (resp. signless Laplacian) spectrum is that Q each eigenvalue ∂iL (resp. ∂i ) does not decrease if an edge is deleted from the graph. To prove the property, we need the following lemma. Lemma 3.4 (Courant–Weyl inequalities, [13]). For a real symmetric matrix M of order n, let λ1 (M ) ≥ λ2 (M ) ≥ · · · ≥ λn (M ) denote its eigenvalues. If N1 and N2 are two real symmetric matrices of order n and if N = N1 + N2 , then for every i = 1, . . . , n, we have

λi (N1 ) + λ1 (N2 ) ≥ λi (N ) ≥ λi (N1 ) + λn (N2 ). Theorem 3.5. Let G be a connected graph on n vertices and m ˜ obtained from G by the deletion of an edge. G 





≥ n edges. Consider the connected graph



• Let ∂1L , ∂2L , . . . , ∂nL and ∂˜1L , ∂˜2L , . . . , ∂˜nL denote the distance Laplacian spectra of G and G˜ respecL ˜L tively.  Then ∂i ≥ ∂i for  all i= 1, . . . , n.  Q Q Q • Let ∂1 , ∂2 , . . . , ∂n and ∂˜1Q , ∂˜2Q , . . . , ∂˜nQ denote the distance signless Laplacian spectra of G and ˜ respectively. Then ∂˜iQ ≥ ∂iQ for all i = 1, . . . , n. G Proof.

• We write the distance Laplacian matrix of G˜ as D˜ L = DL + M, where M expresses the changes

•

in DL due to the deletion of an edge from G. It is easy to see that M is diagonally dominant with nonnegative diagonal entries. Thus M is a positive semi-definite matrix. Also, it is easy to see that 0 is an eigenvalue of M. Now, the result follows immediately from Lemma 3.4. ˜ as D˜ Q = DQ + M, where M expresses the We write the distance signless Laplacian matrix of G Q changes in D due to the deletion of an edge from G. It is easy to see that M is diagonally dominant with nonnegative (diagonal) entries. Thus M is a positive semi-definite matrix, and therefore, the result follows from Lemma 3.4. 

An immediate consequence of the spectra domination resulting from the deletion of an edge is that the distance (signless) Laplacian of any connected graph dominates that of the complete graph of the same order. Corollary 3.6. Let G be a connected graph on n

≥ 3 vertices. Then

• ∂iL (G) ≥ ∂iL (Kn ) = n, for all 1 ≤ i ≤ n − 1, and ∂nL (G) = ∂nL (Kn ) = 0; • ∂1Q (G) ≥ ∂1Q (Kn ) = 2n − 2 and ∂iQ (G) ≥ ∂iQ (Kn ) = n − 2, for all 2 ≤ i ≤ n. Moreover, ∂2Q (G) = ∂2Q (Kn ) = n − 2 if and only if G is the complete graph Kn . Proof. The inequalities follow immediately from Theorem 3.5. Q Q Kn , it suffices to observe To see that ∂2 (G) = ∂2 (Kn ) = n − 2 if and   only if G is the complete graph that if G

Q Q ∼ = Kn , then ∂2 (G) ≥ ∂2 (Kn − e) = 3n − 2 −

√

n2

− 4n + 20 /2 > n − 2. 

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

31

4. Similarities with the algebraic connectivity The study of the second smallest distance Laplacian eigenvalue

∂nL−1 (G)

∂nL−1 of G led to the observation

that = n if and only if G is disconnected. The observation is similar to the well known result [19] about the algebraic connectivity (second smallest Laplacian eigenvalue): μn−1 = 0 if and only if G is disconnected. In fact, the second smallest distance Laplacian eigenvalue ∂nL−1 of G is for G exactly

what the second smallest Laplacian eigenvalue of G is for G, i.e., ∂nL−1 of G can be seen as the algebraic connectivity of G. Before the statement of the main result recall the following lemma due to Fiedler [19]. Lemma 4.1 [19]. If G is a graph on n vertices, then

μn−1 (G) + μ1 (G) = n. Theorem 4.2. Let G be a connected graph on n vertices. Then ∂nL−1

≥ n with equality holding if and only

if G is disconnected. Furthermore, the multiplicity of n as an eigenvalue of DL is one less than the number of components of G.

Proof. First, from Corollary 3.6, for any connected graph G,

∂nL−1 ≥ ∂nL−1 (Kn ) = n. If G is disconnected, the algebraic connectivity of G is μn−1 (G) = 0, thus by Lemma 4.1, n is a Laplacian eigenvalue of G. Also, since G is disconnected, the diameter of G is at most 2 and therefore by Theorem 3.1, ∂nL−1 = n. If G is connected, since adding edges does not increase the eigenvalues of DL (according to Theorem 3.5), it suffices to prove that ∂nL−1  = n when G is a tree. Assume that G is a tree of diameter D. Since G is connected, we have D ≥ 3. If D ≥ 4, then the diameter of G is D = 2 and by Lemma 4.1 and Theorem 3.1, n is not a distance Laplacian eigenvalue of G as the algebraic connectivity of G is not 0. Now, assume that D = 3. All the vertices, but two denoted u and v, are pendent in G. Under these conditions, dG (u, v) = 3 and {u, v} is the only pair of vertices at distance 3 in G. Let d(u) = k and d(v) = l (note that k + l = n − 2) and label the vertices of G v1 , v2 , . . . , vn such that v1 , . . . , vk are the neighbors of u, vk+1 , . . . , vk+l are the neighbors of v, u = vn−1 and v = vn . Using that labeling, we can write the value of the characteristic polynomial of DL at n as follows. ⎤⎞ ⎛⎡ M N ⎦⎠ , PL (n) = det ⎝⎣ NT R

where M

= Jn−2 − In−2 , ⎡

N

=

⎤ k times l times       ⎢ ⎥ ⎢1 1 ... 1 2 2 ... 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 2 2 ... 2 1 1 ... 1

⎡ and

R

=⎣

⎤

−l − 1

3

3

−k − 1

⎦.

The determinant of M is det (M ) = (−1)(n−1) · (n − 3), and the inverse of M is (n − 3)−1 · M , where M is the (n − 2) × (n − 2)-matrix, all diagonal entries of which are equal to 4 − n and all nondiagonal entries are equal to 1. Now, using the properties of the determinants (see for example [13, Lemma 2.2.])

32

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

PL (n) = det (M ) · det (R − N T M −1 N )

= (−1)(n−1) · (n − 3) ⎛⎡

· det ⎝⎣

⎤

−l − 1

3

3

−k − 1

⎦−

⎡ 1 n−3

·⎣

4l − k(l − 1) kl + 2(k + l) kl + 2(k + l) 4k − l(k − 1)

⎤⎞ ⎦⎠

= (−1)(n−1) · 2(n − 3) = 0. Thus n is not an eigenvalue of DL . Note that we used the MAPLE software to evaluate the determinant. From the above lines, if n is an eigenvalue of DL , the diameter of G is necessarily D = 2. Thus the relation between the multiplicity of n as an eigenvalue of DL and the number of components of G follows from Theorem 3.1.  As immediate consequences of the above theorem, we have the following corollaries. Corollary 4.3. Let G be a connected graph on n vertices. Then ∂1L (G) the complete graph Kn .

≥ n with equality if and only if G is

Proof. The bound follows from Theorem 3.5. Let G be a connected graph with ∂1L (G) = n. Then n is an eigenvalue of DL of multiplicity n − 1. Thus G has n components that are necessarily isolated vertices and therefore G is the complete graph.  Corollary 4.4. If G is bipartite and n is among its distance Laplacian eigenvalues, then G is complete bipartite. Therefore, the star Sn is the only tree for which n is a distance Laplacian eigenvalue. Proof. The only bipartite graphs with disconnected complement are the complete bipartite graphs.  Corollary 4.5. If G is a graph with maximum degree  = n − 1, then n is an eigenvalue of DL with multiplicity n − 1 if G is complete, and at least n if G is not complete, where n denotes the number of vertices of maximum degree in G. Proof. It suffices to note that each dominating vertex (a vertex of degree n − 1) of G corresponds to an isolated vertex, and thus to a component, of its complement G.  References [1] M. Aouchiche, G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 8. Variations on Graffiti 105, Congr. Numer. 148 (2001) 129–144. [2] M. Aouchiche, P. Hansen, On a conjecture about the Szeged index, European J. Combin. 31 (2010) 1662–1666. [3] G.A. Baker, Drum shapes and isospectral graphs, J. Math. Phys. 7 (1966) 2238–2242. [4] R.B. Bapat, Determinant of the distance matrix of a tree with matrix weights, Linear Algebra Appl. 416 (2006) 2–7. [5] R.B. Bapat, S.J. Kirkland, M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193–209. [6] R.B. Bapat, A.K. Lal, S. Pati, A q-analogue of the distance matrix of a tree, Linear Algebra Appl. 416 (2006) 799–814. [7] A.E. Brouwer, E. Spence, Cospectral graphs on 12 vertices, Electron. J. Combin. 16 (2009), Paper 20. [8] S. Butler, J. Grout, A construction of cospectral graphs for the normalized Laplacian, Electron. J. Combin. 18 (2011), Paper 231. [9] S. Cabello, P. Lukšic, ´ The complexity of obtaining a distance-balanced graph, Electron. J. Combin. 18 (1) (2011), Paper 49. [10] F.N.K. Chung, Spectral Graph Theory, CBMS Lecture Notes, AMS, Providence, RI, 1997. [11] L. Collatz, U. Sinogowitz, Spektren endlicher grafen, Abh. Math. Sem. Univ. Hamburg 21 (1957) 63–77. [12] D.M. Cvetkovic, ´ Graphs and their spectra, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 354–356 (1971) 1–50. [13] D. Cvetkovic, ´ M. Doob, H. Sachs, Spectra of Graphs—Theory and Applications, third ed., Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995. [14] D. Cvetkovic, ´ P. Rowlinson, S. Simic, ´ Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd) 81 (95) (2007) 11–27. [15] D. Cvetkovic, ´ S.K. Simic, ´ Towards a spectral theory of graphs based on the signless Laplacian. I, Publ. Inst. Math. (Beograd) 85 (99) (2009) 19–33. [16] D. Cvetkovic, ´ S.K. Simic, ´ Towards a spectral theory of graphs based on the signless Laplacian. II, Linear Algebra Appl. 432 (2010) 2257–2272. [17] D. Cvetkovic, ´ S.K. Simic, ´ Towards a spectral theory of graphs based on the signless Laplacian. III, Appl. Anal. Discrete Math. 4 (2010) 156–166.

M. Aouchiche, P. Hansen / Linear Algebra and its Applications 439 (2013) 21–33

33

[18] M. Edelberg, M.R. Garey, R.L. Graham, On the distance matrix of a tree, Discrete Math. 14 (1976) 23–39. [19] M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (98) (1973) 298–305. [20] P.W. Fowler, G. Caporossi, P. Hansen, Distance matrices, wiener indices, and related invariants of fullerenes, J. Phys. Chem. A 105 (2001) 6232–6242. [21] H. Fujii, A. Katsuda, Isospectral graphs and isoperimetric constants, Discrete Math. 207 (1999) 33–52. [22] C.D. Godsil, B. McKay, Products of Graphs and their Spectra, Lecture Notes in Math., vol. 560, Springer, Berlin, 1976, pp. 61–72. [23] C.D. Godsil, B. McKay, Constructing cospectral graphs, Aequationes Math. 25 (1982) 257–268. [24] Hs.H. Günthard, H. Primas, Zusammenhang von graphtheorie und MO—theorie von molekeln mit systemen konjugierter bindungen, Helv. Chim. Acta 39 (1956) 1645–1653. [25] R.L. Graham, L. Lovász, Distance matrix polynomials of trees, Adv. Math. 29 (1978) 60–88. [26] R.L. Graham, H.O. Pollack, On the addressing problem for loop switching, Bell System Tech. J. 50 (1971) 2495–2519. [27] W.H. Haemers, E. Spence, Enumeration of cospectral graphs, European J. Combin. 25 (2004) 199–211. [28] L. Halbeisen, N. Hungerbhler, Generation of isospectral graphs, J. Graph Theory 31 (1999) 255–265. [29] K. Handa, Bipartite graphs with balanced (a, b)-partitions, Ars Combin. 51 (1999) 113–119. [30] H. Hosoya, M. Murakami, M. Gotoh, Distance polynomial and characterization of a graph, Natur. Sci. Rep. Ochanomizu Univ. 24 (1973) 27–34. [31] C.R. Johnson, M. Newman, A note on cospectral graphs, J. Combin. Theory Ser. B 28 (1980) 96–103. [32] B.D. McKay, On the spectral characterisation of trees, Ars Combin. 3 (1977) 219–232. [33] R. Merris, Large families of Laplacian isospectral graphs, Linear and Multilinear Algebra 43 (1997) 201–205. [34] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197/198 (1994) 143–176. [35] R. Merris, The distance spectrum of a tree, J. Graph Theory 14 (1990) 365–369. [36] Z. Mihalic, ´ D. Veljan, D. Amic, ´ S. Nikolc, ´ D. Plavšic, ´ N. Trinajstic, ´ The distance matrix in chemistry, J. Math. Chem. 11 (1992) 223–258. [37] B. Mohar, Graph Laplacians, in: L.W. Beineke, R.J. Wilson (Eds.), Topics in Algebraic Graph Theory, Cambridge University Press, 2004, pp. 113–136. [38] A.J. Schwenk, Almost all trees are cospectral, in: F. Harary (Ed.), New Directions in the Theory of Graphs, Academic Press, New York, 1973, pp. 275–307. [39] J. Tan, On isospectral graphs, Interdiscip. Inform. Sci. 4 (1998) 117–124. [40] E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003) 241–272. [41] E.R. van Dam, W.H. Haemers, J.H. Koolen, Cospectral graphs and the generalized adjacency matrix, Linear Algebra Appl. 423 (2007) 33–41.