Extending a theorem by Fiedler and applicatications

Extending a theorem by Fiedler and applicatications to graph energy María Robbiano Departamento de Matemáticas Universidad Católica del Norte Avenida Angamos 0610 Antofagasta, Chile [email protected] Enide Andrade Martins Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal. [email protected]

Ivan Gutman University of Kragujevac, Kragujevac, Serbia. [email protected]

April 10, 2011

Abstract We use a lemma due to Fiedler to obtain eigenspaces of some graphs and we apply these results to graph energy..

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Notations and preliminaries

Let A be an n n matrix.The scalars and vectors v 6= 0 satisfying Av = v we call eigenvalues and eigenvectors of A; respectively and, any such pair, ( ; v); is called an eigenpair for A: The set of distinct eigenvalues (including multiplicies), denoted by (A); is called the spectrum of A: The eigenvectors of the adjacency matrix of a graph G, together with the eigenvalues, provide a useful tool in the investigation of the structure of the graph. In this work, having a result presented in Research supported by Centre for Research on Optimization and Control (CEOC) from the "Fundação para a Ciência e a Tecnologia - FCT", co…nanced by the European Community Fund FEDER/POCI 2010.

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[1] as motivation, we obtain, for graphs with a special type of adjacency matrix, a lower bound for the energy of these graphs. In 1974 Fiedler obtained the following result [1] . Let A, B be m m and n n symmetric matrices with corresponding eigenpairs ( i ; ui ) ; i = 1; : : : ; m, ( l ; vl ) ; l = 1; : : : ; n; respectively. Lemma 1 [1] Let A, B be m m and n n symmetric matrices with corresponding eigenpairs ( i ; ui ) ; i = 1; : : : ; m, ( i ; vi ) ; i = 1; : : : ; n; respectively. Suppose that ku1 k = 1 = kv1 k : Then, for any the matrix C= has eigenvalues of

2; : : : ;

n;

A v1 uT1

2; : : : ;

m;

b= C

1

u1 v1T B

1;

2,

where

1;

2

are eigenvalues

: 1

We now o¤er a generalization of Fiedler’s lemma. Suppose now that u1 ; : : : ; um (resp. v1 ; : : : ; vn ) constitute an orthonormal system of eigenvectors of A (resp. B). Let ( u1 j ; : : : ; jum ) and ( v1 j ; : : : ; jvn ) be the matrices whose columns consist of the cordinates of the eigenvectors ui and vi associated to the eigenvalues i and i ; respectively. 0 1 0 1 u11 u12 u1m u1p B u21 u22 B u2p C u2m C B C B C In fact, ( u1 j ; : : : ; jum ) = B . ; where, up = B . C C . . . . . . @ . @ . A . . A um1 um2 umm ump for p = 1; : : : ; m and0 1 0 1 v11 v12 v1n v1p B v21 v22 C B C v 2n C B B v2p C ( v1 j ; : : : ; jvn ) = B . , where vp = B . C for p = 1; : : : ; n: C . . .. .. A @ .. @ .. A vn1 vn2 vnn vnp

Lemma 2 Let k min fm; ng and U = ( u1 j ; : : : ; juk ) and V = ( v1 j ; : : : ; jvk ) : Then, for any ; the matrix A UV T C= T VU B has eigenvalues k+1 ; : : : ; m ; k+1 ; : : : ; n ; 1j ; 2j ; j = 1; 2; : : : ; k; where for s = 1; 2; sj is an eigenvalue of bj = C

j

: j

2

Proof. For j = 1; 2; : : : ; k let sj ; w csj ; s = 1; 2 be an eigenpair of the matrix j bj = C ; j

where w bsj = w1sj

w2sj

T

: Then, for j = 1; 2; : : : ; k; s = 1; 2; we have, w1sj w2sj

j j

let

w1sj uj w2sj vj

=

w1sj : w2sj

sj

be an (m + n) vector :Then,

A V UT

UV T B

w1sj uj w2sj vj

w1sj Auj + w2sj U V T vj w1sj V U T uj +w2sj Bv j

=

as

0 1 0 u1j B .. C B.C B u2j C B C B C T C w1sj Auj + w2sj U V vj = w1sj j B .. C + w2sj U B B1C and @ . A B .. C @.A umj 0 0 1 0 0 1 v1j B .. C B.C B v2j C B C B C C w1sj V U T uj +w2sj Bv j = w1sj V B . C: B1C + w2sj j B @ .. A B .. C @.A vnj 0 Therefore, 0

A UV T w1sj uj T w2sj vj VU B (w1sj j + w2sj ) uj = w1sj + w2sj j vj

1

w1sj j uj + w2sj uj w1sj vj +w2sj j vj sj w1sj uj sj w2sj vj w1sj uj : sj w2sj vj

= = =

Recall that

csj sj ; w

; s = 1; 2 is an eigenpair of bj = C

j

; j

3

where w bsj = w1sj

w2sj

T

: Then, for s = 1; 2; we have,

j j

w1sj w2sj

sj

w1sj : w2sj

sj ;

w1sj uj w2sj vj

=

Therefore, for j = 1; 2; : : : ; k; s = 1; 2;

are 2k eigenpairs

for C: For i = k + 1; : : : ; m; we have A V UT

UV T B

ui 0

=

Aui 0

=

0 vt

=

0 Bvt

=

i

ui 0

and for t = k + 1; : : : ; n we set A V UT

UV T B

ui 0 eigenpairs for C: Thus the result is proved. Therefore, for i = k+1; : : : ; m

2

i;

t

0 : vt

and for t = k+1; : : : ; n

t;

0 vt

Applications

A simple graph G is a pair of sets (V; E) such that V is a nonempty …nite set of n vertices and E is the set of m edges. We say that G is a simple (n; m) graph. Let A(G) be the adjacency matrix of the graph G. Its eigenvalues 1 ; : : : ; n form the spectrum of G. (cf. [2]). The notion of energy of an (n; m) graph G (written E(G)) is a nowadays much studied spectral invariant. This concept is of great interest in a vast range of …elds, aspecially in chemistry since it can be used to approximate the total -electron energy of a molecule (cf. [3], [4], [5] and [6]). It is de…ned by n X E(G) = j jj : j=1

Given a complex m n matrix C; we index its singular values by s1 (C); s2 (C); : : : : The value X E (C) = sj (C); j

is the energy of C (cf:[10]), thereby extending the concept of graph en4

are

ergy. Consequently, if the matrix C 2 Rn 1 (C); : : : ; n (C); its energy is given by E (C) =

n X i=1

n

is symmetric with eigenvalues

j i (C)j :

In this section, using Lemma 2, we present an application of the concept of energy to some special kinds of graphs. We …rst formulate an auxiliary result. Let Q = (qij ) be an n1 n2 matrix with real-valued elements. Without loss of generality we may assume n1 n2 : Let the singular values of Q Pthat n1 be s1 ; s2 ; :::; sn1 : Then E(Q) = i=1 si :

Lemma 3

E(Q)

p

n1 kQkF

where kQkF is the Frobenius norm of Q; de…ned as, v uX n2 u n1 X 2: t kQkF = qij i=1 j=1

[17].

Proof. Let

1 p

1 ; 2 ; :::; p

p X

be real numbers. Their variance v i=1 u p p X X p u 2 pt is known to be non-negative. Therefore, i i. i=1

2 i

1 p

p X i=1

i

!2

i=1

Setting in the above inequality p = n1 and i = si ; we obtain v u n1 n1 X X p u n1 t s2i : E(Q) = si i=1

i=1

Now, the singularities of the matrix Q are just the square roots of the n1 X T eigenvalues of QQ . Therefore, s2i is equal to the sum of the eigenvalues i=1

of QQT , which in turn is the trace of QQT . Lemma 3 follows from T r(QQT ) =

n1 X n2 X

T qij qji =

i=1 j=1

n1 X n2 X i=1 j=1

5

2 qij = kQk2F .

Let B XT

A(G) =

X C

be a partition of the adjacency matrix of a graph G; where B and C have orders n1 n1 and n2 n2 ; respectively. Thus X has order n1 n2 : Morever, let k min fn1 ; n2 g and consider the eigenpairs f( i ; ui ) : 1 i kg of B f( i ; vi ) : 1 i kg of C such that the sets fu1 ; : : : ; uk g and fv1 ; : : : ; vk g are orthonormal vectors. Let U = ( u1 j : : : juk ) and V = ( v1 j : : : jvk ). Theorem 4 Let G be a graph of order n = n1 + n2 , such that its adjacency B X , where B and C represent adjacency matrix is given by A(G) = XT C matrices of graphs. Morever, let k min fn1 ; n2 g and consider the eigenpairs f( i ; ui ) : 1 i kg of B f( i ; vi ) : 1 i kg of C such that the sets fu1 ; : : : ; uk g and fv1 ; : : : ; vk g are orthonormal vectors. Let U = ( u1 j : : : juk ) and V = ( v1 j : : : jvk ). Then 1 2

E(G) p

n1

p

k

k P

i

i=1

s

kBk2F

+

i k P

+

i=1

q

(

2

i)

i

j i j2 +

p

n2

+ 12

+4

k

s

k P

i

+

i=1

kCk2F

k P

i=1

q (

i

2 i)

i

+4 +

j i j2 +

min fn1 ; n2 g X U V T F : 0 X UV T B UV T B X = + = Proof. A(G) = XT V U T 0 V UT C XT C M + N: Thus, according by Ky Fan theorem [16], E(G) E(M ) + E(N ): 1 i ci = ci = is M Note that the spectrum of the matrix M 1 i q 2 1 ( i : Then appling Lemma 2, i+ i i) + 4 2 2

E(M ) =

q 1 Pk 1 Pk 2 + + ( i i i i) + 4 + i=1 2 2 i=1 Pn1 Pn2 + i=k+1 j i j + i=k+1 j i j :

By Lemma 3; p Pn1 Pn2 n1 i=k+1 j i j+ i=k+1 j i j q p Pk 2 + n2 k kCk2F i=1 j i j :

k

6

qP

n1 2 i=k+1 j i j

=

i+

p

n1

q

i

k

q

(

i

kBk2F

2 i)

Pk

+4

2 i=1 j i j

Moreover, also by Lemma 3, E(N ) = 2E(X U V T )

2

p

min fn1 ; n2 g X

UV T

Consider the following special case of Theorem 4. Let both submatrices B abd C be equal, say equal to A, the adjacency matrix of an (n; m)-graph G .Further, let k = n1 = n2 = n: By setting in Lemma 2; A = B = A(G); and considering ( i ; ui ) ; i = 1; : : : ; n, such that U = ( u1 j ; : : : ; jun ) is an orthonormal matrix and = 1; the matrix C=

UUT A

A UUT

has eigenvalues s1 ; s2 ; : : : ; 1; 2; sj is an eigenvalue of

sn ,

A In ; In A

=

s = 1; 2; where for j = 1; 2; : : : ; n; s =

bj = C

j

1

1

j

;

respectively. Thus 1j = j 1; 2j = j + 1 and for the symmetric matrix C; we have P P P P E(C) = ni=1 j i 1j+ ni=1 j i + 1j j ni=1 ( i 1)j+j ni=1 ( i + 1)j = n+n = 2n:

In conclusion, the energy of the graph whose adjacency matrix is C is greater than its number of vertices. At this point it is worth noting that the graph whose adjacency matrix is C is just the sum of the graphs G and K2 , denoted by G + K2 , where K2 is the complete graph on two vertices. The graph operation marked by + is described in detail in textbooks [quote Cvetkovic]. For the present consideration it is important that the spectrum of G1 + G2 consists of the sums of eigenvalues of G1 and G2 [2]. In view of this, the …nding that 1j = j 1; 2j = j +1 is an immediate consequence of the fact that the spectrum of K2 consists of the numbers +1 and 1: The result E(C) 2n can now be generalized as follows. Let 1 ; :::; n1 and 1 ; :::; n2 be, respectively, the eigenvalues of G1 and G2 . Then the eigenvaues of G1 +G2 are of the form i + j , i = 1; :::; n1 ; j = n1 P n2 P 1; :::; n2 ; and E(G1 + G2 ) = i + j : We have, i=1 j=1

E(G1 + G2 )

n2 n1 P P

i=1 j=1

i

+

j

=

n1 P

i=1

7

jn2 i j = n2

n1 P

i=1

j i j = n2 E(G1 )

F

:

because of

n2 P

i

= n2

i

and

j=1

n2 P

j=1

j

= 0:

In an analogous manner it can be shown that E(G1 + G2 ) n1 E(G2 ): The sum G1 + G2 has n1 n2 vertices. Bearing this in mind, we see that if either the energy of G1 exceeds or is equal to the number of vertices of G1 or the energy of G2 exceeds or is equal to the number of vertices of G2 , then the energy of G1 + G2 exceeds the number of vertices of G1 + G2 : Let A; B; C; X; U and V be the same matrices as in the formulation B UV T and proof of Theorem 4. Let, as before M = and Q = V UT C 0 UV T : V UT 0 Theorem 5 E(A)

E(M )

b= Proof. Let A

B XT

E(Q): X C

b We ; then as well known E(A) = E(A):

have

b = M + Pb A = M + P and A

where P =

0 XT

X V

UT

UV T 0

Therefore By Ky Fan theorem [16],

b = 2M A+ A

2E(M )

0

and Pb =

XT

UV T

X V

UT

0

:

2Q:

b + 2E(Q) E(A) + E(A)

implying the theorem. As a special case of Theorem 5, for k = 1, M=

B v1 uT1

u1 v1T C

and Q =

Hence, E(Q) = 2 v1T u1 and E(A)

1 2

By Fiedler’s Lemma 1, E(M ) = q 2 ( 1 (j 1+ 1 1) + 4

E(M )

8

+j

1 j):

:

2 v1T u1 :

E(B)+E(C)+ 21 1j

u1 v1T 0

0 v1 uT1

1

+

1

+

q

(

1

2 1)

+4 +

Therefore E(A) " =

1 2

E(B) + E(C) + ";where q 2 1 ( 1 1+ 1+ 1+ 1) + 4 + 2

q

1

(

1

2 1)

+4

+ j 1 j) 2 v1T u1 : The inequality E(A) E(B) + E(C) has been reported earlier [18]. Therefore our result will be an improvement of that inequality only if " > 0: q

(j

1j

If

1

1 2

1

> 0,

+

1

> 0 and

1

q

(

1

2 1)

1 1

+4

1; then

(j

1 2

1 j + j 1 j)

1

+

1

+

(

1

2 1)

+4 +

= 0 and therefore " < 0: Thus,

in order to get an improvement, we must choose 1 and 1 such that 1 1 < 1: For instance, for 1 = 1 = 0; " = 2 2 v1T u1 which is positive because kv1 k ku1 k = 1 1 = 1: of v1T u1

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[9] P. Lancaster, M.Tismenetsky. The theory of matrices. Academic Press. INC. [10] V. Nikiforov, The energy of graphs and matrices, Journal of Mathematical Analysis and Applications 326, Issue 2 (2007), pp. 1472-1475. [11] O. Rojo and R. Soto. The spectra of the adjacency matrix and Laplacian matrix for some balanced trees. Linear Algebra Appl. 403 (2005) 97-117. [12] O. Rojo. The spectra of a graph obtained from copies of a generalized Bethe tree. Linear Algebra and its Applications 420, I. 2-3 15 (2007) pp. 490-507. [13] O. Rojo. The spectra of some trees and bounds for the largest eigenvalues of any tree. Linear Algebra Appl. 414 (2006) 199-217. [14] R. S. Varga. Matrix Iterative Analysis. Prentice-Hall, Inc., 1965. [15] O. Rojo and M. Robbiano. On the spectra of some weighted rooted tree and applications, Linear Algebra Appl. 420 (2007) 310-328. [16] W. So, M. Robbiano, Nair Maria Maia de Abreu and Ivan Gutman, Applications of the Ky Fan theorem in the theory of graph energy. Lin. Algebra Appl. Linear Algebra Appl. (2009) (IN PRESS). [17] Carl D. Meyer. Matrix Analysis and Applied Linear Algebra. Siam. (2005). [18] Jane Day, W. So. Graph energy change due to edge deletion. Lin. Algebra Appl. Linear Algebra Appl. 428 (2007) 2070-2078.

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