Characteristics polynomial of normalized Laplacian for trees

Applied Mathematics and Computation 271 (2015) 838–844

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Characteristics polynomial of normalized Laplacian for trees Anirban Banerjee a,b,∗, Ranjit Mehatari a a b

Department of Mathematics and Statistics Department of Biological Sciences, Indian Institute of Science Education and Research, Mohanpur Kolkata 741246, India

a r t i c l e

i n f o

a b s t r a c t

MSC: 05C50 05C05

Here, we find the characteristics polynomial of normalized Laplacian of a tree. The coefficients of this polynomial are expressed by the higher order general Randic´ indices for matching, whose values depend on the structure of the tree. We also find the expression of these indices for starlike tree and a double-starlike tree, Hm (p, q). Moreover, we show that two cospectral Hm (p, q) of the same diameter are isomorphic.

Keywords: Normalized Laplacian Characteristics polynomial Tree Starlike tree Double Starlike tree Randic´ index

© 2015 Elsevier Inc. All rights reserved.

1. Introduction Let  = (V, E ) be a simple finite undirected graph of order n. Two vertices u, v ∈ V are called neighbors, u ∼ v, if they are connected by an edge in E, u  v otherwise. Let dv be the degree of a vertex v ∈ V, that is, the number of neighbors of v. The normalized Laplacian matrix [7], L, of  is defined as:

⎧ 1 if u = v and dv = 0, ⎪ ⎨ 1 L( )u,v = − √d d if u ∼ v, u v ⎪ ⎩

(1)

0 otherwise.

This L is similar to the normalized Laplacian  defined in [2,20]. Let φ (x) = det (xI − L) be the characteristics polynomial of L( ). Let us consider φ (x) = a0 xn − a1 xn−1 + a2 xn−2 − · · · + (−1)n−1 an−1 x + (−1)n an . Now if  has no isolated vertices then  ) − i∼ j d 1d and an = 0 (for some properties of φ (x) see [8]). The zeros of φ (x) are the eigenvalues a0 = 1, a1 = n, a2 = n(n−1 2 i j

of L and we order them as λ1 ≥ λ2 ≥ · · · ≥ λn = 0.  is connected iff λn−1 > 0. λ1 ≤ 2, the equality holds iff  has a bipartite component. Moreover,  is bipartite iff for each λi , the value 2 − λi is also an eigenvalue of  . See [7] for more properties of the normalized Laplacian eigenvalues.

1.1. General Randi´c index for matching There are different topological indices. The degree based topological indices [11,21], which include Randi´c index [23], reciprocal Randi´c index [16], general Randi´c index [4], higher order connectivity index [3,22,24], connective eccentricity index [29,30], Zagreb index [1,9,12–14,18,27,28] are more popular than others. ∗

Corresponding author. Tel.: +91 9775320480; fax: +91 3325873020. E-mail addresses: [email protected] (A. Banerjee), [email protected] (R. Mehatari).

http://dx.doi.org/10.1016/j.amc.2015.09.054 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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For any real number α , the general Randic´ index of a graph  is defined by Bollobás and Erdös (see [4]) as:



Rα ( ) =

(di d j )α ,

(2)

i∼ j

which is the general expression of the Randic´ index (also known as connectivity index) introduced by Randic´ in 1975 [23] by choosing α = −1/2 in (2). For more properties of the general Randic´ index of graphs, we refer to [5,16,17,24–26]. The Zagreb index of a graph was first introduced by Gutman and Trinajstic´ [12] in 1972. For a graph  the first and the second Zagreb indices are defined by



Z1 ( ) =



di2 and Z2 ( ) =

i∈V

di d j ,

i∼ j

respectively. Now, for any positive integer p, we define the pth order general Randi´c index for matching as



Rα( p) ( ) =



s(e)α ,

(3)

M p ∈M p ( ) e∈M p

where s(e) = du dv is the strength of the edge e = uv ∈ E, M p is a p-matching, that is, a set of p non-adjacent edges and M p ( ) is the set of all p-matchings in  . The first order general Randic´ index for matching with α = 1 is the second Zagreb index, that is, (1) (1) Z2 ( ) = R1 ( ). We take Rα ( ) = Rα ( ) and

Rα(0) ( ) =



0 if  is the null graph, 1 otherwise. (i)

(2)

(2)

(2)

If  is r-regular, then Rα = r2iα | Mi ( ) |. The R−1 for some known graphs are as follows: R−1 (Sn ) = 0, R−1 (Pn ) = (2) n2 −n−4 , R−1 (Cn ) 32

=

(2) n(n−3) 32 , R−1 (K p,q )

=

( p−1)(q−1) 4pq

(2)

, and R−1 (Kn ) =

usual meanings.

3(n4) , (n−1)4

where the notations, Sn , Pn , Cn , Kn and Kp,q have their

Theorem 1.1. For any real number α ,

Proof.

2



1 Rα ( ) 2

0 ≤ R(α2) ( ) ≤

[Rα ( )] = 2



−

1 R2α ( ) 2

2 (s(e))α

e∈E

=





(s(e))2α + 2

(s(e1 )s(e2 ))α + 2

e1 ,e2 ∈M2 ( )

e∈E



(s(e1 )s(e2 ))α

e1 ,e2 ∈M / 2 ( )

which proves our required result.  ( p)

( p)

( p)

Clearly, for any two graphs  1 ,  2 , and p ≥ 0, Rα (1 ∪ 2 ) ≥ Rα (1 ) + Rα (2 ). The equality holds, when p = 1 or one of the graphs is null. It has been seen that the matching plays a role in the spectrum of a tree. In [6], some results on normalized Laplacian spectrum for trees have been discussed. Now, we derive (or express the coefficients of) the characteristics polynomial φ T (x) of a tree T in (i) terms of R−1 (T ). 2. The characteristics polynomial of normalized Laplacian for a tree

Theorem 2.1. Let T be a tree with n vertices and maximum matching number k, then

φT (x) =

k 

i) (−1)i (x − 1)n−2i R(−1 (T )

(4)

i=0

and the coefficients of φ T are given by

ap =

k  i=0



(−1)i



n − 2i (i) R (T ). p − 2i −1

(5)

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A. Banerjee, R. Mehatari / Applied Mathematics and Computation 271 (2015) 838–844

V

Fig. 1. Starlike tree, T(1, 2, 2, 3, 4).

Proof. Consider a matrix, B = [bi j ] = xIn − L, where

bi j =

⎧ x − 1 if i = j ⎪ ⎨ ⎪ ⎩

√1

if i ∼ j

di d j

0 else.

Now,

φT (x) = det (B) =



σ ∈Sn

bσ ,

where bσ = sgn(σ )b1,σ (1) b2,σ (2) . . . bn,σ (n) and Sn is the set of all permutation of {1, . . . , n}. Now, for any σ ∈ Sn and σ (i) = i, bσ = 0 only when i ∼ σ (i). Since, T does not contain any cycle, here, σ is either the identity permutation or a product of disjoint transpositions. When σ is the identity permutation, bσ = (x − 1)n and if σ = (i1 σ (i1 ))(i2 σ (i2 )) . . . (il σ (il )) is a product of disjoint transpositions, then,



bσ =

(−1)l (x − 1)n−2l di

1

0 otherwise.

1 1 dσ (i ) di2 dσ (i ) 1 2

···

1 di dσ (i ) l l

if i j ∼ σ (i j ) ∀ j

Now, since, T has the maximum matching number k,

φT (x) =

k  

(−1)|M| (x − 1)n−2|M|

i=0 M∈Mi

=

k 

 1 s(e)

e∈M

i) (−1)i (x − 1)n−2i R(−1 (T ).

i=0

Expanding the right hand side of the above equation we get

ap =

k  i=0



(−1)i



n − 2i (i) R (T ). p − 2i −1 

Corollary 2.1. For a tree T with maximum matching number k, 2) k) 1 − R−1 (T ) + R(−1 (T ) − · · · + (−1)k R(−1 (T ) = 0.

Corollary 2.2. Let T be a tree with maximum matching number k. The eigenvalues of T are 1 with the multiplicity n − 2k, and 1 ± (1 ≤ i ≤ k) where α i ’s are the zeros of the polynomial k) ψT (y) = yk − R−1 (T )yk−1 + · · · + (−1)k R(−1 (T ).

(6) √

αi

(7) (i)

The characteristics polynomial φ T (x) of a tree T can be expressed in terms of R−1 (T ), whose value depends on the structure (i)

of T. Now, we find the expression of R−1 (T ) for two different trees, starlike tree [10,22] and a specific type of double starlike trees [15,19]. 2.1. Starlike tree A tree is called starlike tree (see Fig. 1) if it has exactly one vertex v of degree grater than two. We denote a starlike tree with dv = r, 3 ≤ r ≤ n − 1, by T (l1 , l2 , . . . , lr ) where li ’s are positive integers with l1 + l2 + · · · + lr = n − 1, that is, T (l1 , l2 , . . . , lr ) − v =

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Pl1 ∪ Pl2 ∪ · · · ∪ Plr where Pli is, a path on li vertices, connected to v (see Fig. 1 for an example). Now onwards, without loss of any generality, we assume 1 ≤ l1 ≤ l2 ≤ · · · ≤ lr . Theorem 2.2. Let T (l1 , l2 , . . . , lr ) be a starlike tree on n vertices with maximum matching number k. If there are exactly m (m = 0) number of li ’s which are odd numbers then , (i) T has the maximum matching number n−m+1 2 1 m 2 (ii) R−1 (T ) = (n + 1) + ( − 1). 4 4 r Furthermore, if l p1 , l p2 , . . . l pm are odd then  m (l p j + 1)  1 (iii) |Mk (T )| = , and . m−1 lp j + 1 2 j=1  m lp j  1 (k) (iv) R−1 (T ) =  . . lp j di j=1

Proof. (i) This part is obvious, since a maximal matching can be taken as follows: for each even li , we cover the corresponding Pli by a perfect matching. Consider another perfect matching on a Pli +1 where v ∈ Pli +1 and li is odd. For all other odd li ’s, take li −1 2

matching. (ii) T has m edges of strength r, (r − m) edges of strength 2r, (r − m) edges of strength 2 and the rest (n − 2r + m − 1) edges are of strength 4. Thus,

R−1 (T ) =

 1 s(e) e∈E

=

1 m 2 (n + 1) + ( − 1). 4 4 r

(iii) Px has maximum matching number

|Mk (T )| =

x−1 2

with

x+1 2

number of matchings, when x is odd. Thus,

m m   (l pk + 1) 2 j=1 k=1 k= j

 =

m (l p j + 1) 

2m−1

.

j=1

1 . lp j + 1

(iv) To get a maximal matching in T (l1 , l2 , . . . , lr ), where l p1 , l p2 , . . . , l pm are odd, one l p j is combined with v1 and the in other odd l p j one vertex, of degree one or two, remains uncovered. One or vertex is one or two respectively. Thus,

l p −1 j

2

positions are possible if the degree of the uncover

⎡

1 ⎢ k) R(−1 (T ) =  ⎣m + (m − 1).2 di

m  j=1

(l p j − 1) 2

⎤

⎡

+ · · · + 2k−1

⎤

m m   j=1 k=1 k= j

(l pk − 1) ⎥ ⎦ 2

m m 1 ⎢  ⎥ =  ⎣ (1 + (l pk − 1))⎦ di

 = 

j=1 k=1 k= j

m lp j  1 . . lp j di j=1

 Corollary 2.3. If T is a tree as in Theorem (2.2), then the multiplicity of the eigenvalue 1 is m − 1. Remark. If m = 0 in Theorem (2.2), then T has maximum matching number k =

n−1 2

with |Mk (T )| =

n+1 2

(k)

and R−1 (T ) =

n−1  . di

Example. The spectrum of different starlike trees with n = 8 vertices are given bellow. Superscripts in the table show the algebraic multiplicity of an eigenvalue.

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Fig. 2. Double starlike tree H2 (4, 4) and H4 (3, 5).

1.

Partition

Randic´ indices

λ(L)

1,1,5

25 21 11 , , 12 16 48

0,2,12 ,1 ±

√

√ 13± 37 √ 2 6

2.

1,2,4

13 3 17 1 , 2 , 48 , 48 6

0,2,1 ± 0.876,1 ± 0.558,1 ± 0.295

3.

1,3,3

13 71 5 , 48 , 16 6

0,2,12 ,1 ±

4.

2,2,3

9 5 , , 7,1 4 3 16 48

0,2,1 ±

1,1,1,4

6.

1,1,2,3

2, 39 , 32

7.

1,2,2,2

17 3 13 , , , 1 8 2 32 32

0,2,(1 ±

8.

1,1,1,1,3

33 13 , 20 20

0,2,1 ,1 ±

9.

1,1,1,2,2

9 19 , , 3 5 20 20

0,2,12 ,1 ±

10.

1,1,1,1,1,2

17 , 5 12 12

0,2,14 ,1 ±

7 32

3 ,1 2

√ √5 2 3 √ 9± 57 √ 2 6

± √

√1 ,1 ± 2 √ 0,2,1 ,1 ± 23 ,1 ± 2√1 2 √ √ 2 √ 0,2,12 ,1 ± 24± 2

15 31 3 , 32 , 32 8

5.

√

2

) ±  13

√1 2 ,1 2

4

 

1 √ 2 2

20 3 ,1 10

±

√1 2

5 12

2.2. Double starlike tree A tree T is called double starlike if it has exactly two vertices of degree greater than two. Let Hm (p, q) be a double starlike tree obtained by attaching p pendant vertices to one end-vertex of a path Pm and q pendant vertices to the other end-vertex of Pm . Thus, H2 (p, q) is a double star S p+1,q+1 , that is, a tree with exactly two non-pendant vertices with the degree p + 1 and q + 1 respectively. See Fig. 2 for examples. Lemma 2.1. Let Ci,n and k be the number of i-matchings and maximal matching number, respectively, of Pn (n ≥ 3), then

1 1 Ci,n−2 + i−1 Ci−1,n−2 , and 4i 4 k−1  1 ψPn (y) = (y − 1) (−1)i i Ci,n−2 yk−1−i , 4

i) R(−1 (Pn ) =

i=0

where ψPn (y) is defined as in Corollary 2.2. Proof. The maximum matching number of Pn is  n2 and Ci,n = (n−i i ). Pn has n − 3 edges of strength 4 and two edges of strength 2. Thus,

1 1 1 1 Ci,n−2 + 2 × × i−1 Ci−1,n−3 + i−1 Ci−2,n−4 2 4i 4 4 1 1 = i Ci,n−2 + i−1 Ci−1,n−2 . 4 4

i) R(−1 (Pn ) =

which proves the first part of the theorem. (k) For the second part of the lemma, we have, R−1 (Pn ) = Therefore,

1 C . 4k−1 k−1,n−2

1 4

ψPn (y) = (yk − yk−1 ) − C1,n−2 (yk−1 − yk−2 ) + · · · 1 Ck−1,n−2 (y − 1) 4k−1 k−1  1 = (y − 1) (−1)i i Ci,n−2 yk−1−i . 4 +(−1)k−1

i=0

 Theorem 2.3. Let T be the double starlike tree Hm (p, q), then

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843

(i) T has the maximum matching number  m 2 + 1, and the multiplicity of the eigenvalue 1 is p + q − 2 if m is even and p + q − 1 if m is odd, (i) (i) (i−1) (i−1) p+q (ii) R−1 (T ) = R−1 (Pm ) + ( p+1pq )(q+1) R−1 (Pm ) + 2( p+1)(q+1) R−1 (Pm−1 ),

(iii) and

ψT (y) =

(y − r1 )ψPm (y) − r2 ψPm−1 (y) i f m odd, y(ψPm (y) − r2 ψPm−1 (y)) − r1 ψPm (y) i f m even,

(8)

p+q where r1 = ( p+1pq )(q+1) , r2 = 2( p+1)(q+1) and ψ T (y) is defined as in Corollary 2.2.

Proof. (i) This is easy to verify. (ii) When m ≥ 3, T has p edges of strength p + 1, q edges of strength q + 1, 1 edge of strength 2( p + 1), 1 edges of strength 2(q + 1) and rest m − 3 edges of strength 4. Thus,









q 1 1 1 1 p 1 + Ci,m−2 + i−1 Ci−1,m−2 + i−1 + Ci−1,m−3 p+1 q+1 2( p + 1) 2(q + 1) 4i 4 4 1 1 1 pq p+q 1 + i−2 Ci−2,m−2 + i−2 Ci−2,m−3 + i−2 Ci−2,m−4 4 ( p + 1)(q + 1) 4 2( p + 1)(q + 1) 4 4( p + 1)(q + 1) 1 1 1 pq = i Ci,m−2 + i−1 Ci−1,m−2 + i−1 Ci−1,m−2 4 4 4 ( p + 1)(q + 1) 1 1 1 pq p+q p+q + i−2 Ci−2,m−2 + i−1 Ci−1,m−3 + i−2 Ci−2,m−3 4 ( p + 1)(q + 1) 4 2( p + 1)(q + 1) 4 2( p + 1)(q + 1) pq p+q i) = R(−1 (Pm ) + R(i−1) (P ) + R(i−1) (Pm−1 ) ( p + 1)(q + 1) −1 m 2( p + 1)(q + 1) −1

i) R(−1 (T ) =

Again if m = 2, then

R−1 (T ) = 1 + 2) R(−1 (T ) =

pq

( p + 1)(q + 1) pq

( p + 1)(q + 1)

, and

. (k)

(i)

(i)

(i−1)

(i−1)

(iii) We have, ψT (y) = yk − R−1 (T )yk−1 + · · · + (−1)k R−1 (T ) and R−1 (T ) = R−1 (Pm ) + r1 R−1 (Pm ) + r2 R−1 (Pm−1 ), 1 ≤ i ≤ k, where k is the maximum matching number of T. Hence the maximum matching number of Pm is k − 1 and the same is of Pm−1 is k − 1 if m odd and k − 2 otherwise. Thus, (k)

R−1 (T ) =



k−1) k−1) r1 R(−1 (Pm ) + r2 R(−1 (Pm−1 ) (k−1) r1 R−1 (Pm )

if m odd, if m even.

(9)

Hence the result follows.  (1)

Example. Consider the double starlike trees as in Fig. 2. For the first tree, T1 = H2 (4, 4) we have R−1 (T1 ) =

41 25

(2)

and R−1 (T1 ) =

16 25 .

Thus the eigenvalues of T1 are 0, 16 , 2 0.2, 1.8. The second tree, T2 = H4 (3, 5) has maximum matching number equals to 3 and (1) (2) (3) 115 5 6 the Randic´ indices of matching are R−1 (T2 ) = 49 24 , R−1 (T2 ) = 96 and R−1 (T2 ) = 32 . Hence the eigenvalues of T2 are 0, 1 , 2, 1 ± 0.4263, 1 ± 0.9273. Theorem 2.4. Let T1 = Hm ( p1 , q1 ) and T2 = Hm ( p2 , q2 ) be L-cospectral. Then, T1 and T2 are isomorphic.

Proof. From Theorem (2.3) and (2.1) we have, p1 + q1 = p2 + q2 and p1 q1 = p2 q2 . Hence the proof.  3. Summary and conclusions The Zegreb indices and the Randic´ index are of great importance for molecular chemistry. They are used to characterize the molecular branching in chemical graphs. The general Randic´ indices for matching can also play an important role in this area. It can be used to characterize different classes of graphs. The estimation of general Randic´ indices for matching for trees, which are the simplest structure amongst the graphs, is much easier than others. Corollary 2.1 states that, for a n-vertex tree with maximum matching number k it is sufficient to calculate the zeros of a k degree polynomial to determine its complete spectrum. Furthermore, the Corollary 2.1 shows that the general Randic´ indices are related by a simple equation. Thus, we only need to calculate the k − 1 general Randic´ indices for matching to compute the complete set of eigenvalues.

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