On Signed Paths, Signed Cycles and their Energies

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Laplacian matrices of signed paths and signed cycles and thereby their energies ... graph of Γ and σ : E → {+1, −1},is a function called a signature or signing . E.
Applied Mathematical Sciences, Vol. 4, 2010, no. 70, 3455 - 3466

On Signed Paths, Signed Cycles and their Energies Germina K.A. Research Center & PG Department of Mathematics Mary Matha arts & Science College Vemom P.O., Mananthavady - 670645, India [email protected] Shahul Hameed K Department of Mathematics Government Brennen College Thalassery - 670106, Kerala, India [email protected]

Abstract. In this article, we evaluate the eigenvalues of the adjacency and Laplacian matrices of signed paths and signed cycles and thereby their energies are computed. It is shown that all signed paths of order n are equienergetic under the respective matrices. Moreover, recurrence formulas for the characteristic polynomials of signed paths and signed cycles in the case of their adjacency and Laplacian matrices are also established. Mathematics Subject Classification: 05C22 Keywords: signed graph, signed path, signed cycle, difference equation, recurrence formula, adjacency matrix, Laplacian matrix

Introduction All graphs in this article are simple and loop-free. Let G = (V, E) be a simple graph of order n and size m with vertex setV = {v1 , v2 , ...., vn } and edge setE = {e1 , e2 , ...., em }. Let A(G) be the adjacency matrix, L(G) = D(G) − A(G) be the Laplacian matrix, where D(G) is the diagonal matrix of degrees of the vertices and Q(G) = D(G) + A(G) be the signless Laplacian of the graph G.

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Signed graphs (also called sigraphs) are much studied in literature because of their extensive use in modeling a variety of socio-psychological processes and also because of their interesting connections with many classical mathematical systems([7],[11] and [13]). Formally, a sigraph is an ordered pair Γ = (G, σ) where G = (V, E) is a graph called the underlying graph of Γ and σ : E → {+1, −1},is a function called a signature or signing . E + (Γ) denotes the set of all edges of G that are mapped by σ to the element +1 and E − (Γ) = E − E + (Γ). The elements of E + (Γ) are called positive edges and those of E − (Γ) are called negative edges of Γ. A signed graph is all-positive (respectively, all-negative) if all of its edges are positive (negative); further,it is said to be homogeneous if it is either all-positive or all-negative and heterogeneous otherwise. The sign of a cycle in a signed graph is the product of the signs of its edges. Thus a cycle is positive if and only if it contains even number of negative edges. A signed graph Γ is said to be balanced or cycle balanced if all of its cycles are positive.Two signed graphs, Γ1 = (G1 , σ1 ) and Γ2 = (G2 , σ2 ), are isomorphic if there is a graph isomorphism f : G1 → G2 that preserves signs of the edges. If θ : V → {+1, −1} is switching function, then switching of the signed graph Γ = (G, σ) by θ means changing σ to σ θ defined by: σ θ (uv) = θ(u)σ(uv)θ(v). The switched graph denoted by Γθ , is the signed graph Γθ = (G, σ θ ). We call two signed graphs Γ1 = (G, σ1 ) and Γ2 = (G, σ2 ) to be switching equivalent and write Γ1 ∼ Γ2 , if there exists a switching function θ : V → {+1, −1} such that Γ1 = Γθ2 . It can be seen that switching preserves many features of the two signed graphs including the eigenvalues [14]. Two matrices M1 and M2 of order n are said to be signature similar if there exists a diagonal matrix S = diag(s1 , s2 , . . . , sn ) with diagonal entries si = ±1 such that M2 = SM1 S −1 . Observe that S is invertible and S = S −1 . Indeed, from this definition, it is clear that two matrices which are signature similar will have the same eigenvalues. Remark 0.1. From the definition of the signature similar matrices and switching equivalent signed graphs, it can be seen that two signed graphs, Γ1 = (G, σ1 ) and Γ2 = (G, σ2 ) on the same underlying graph G, are switching equivalent if and only if their adjacency matrices are signature similar. Observe that switching equivalence of Γ1 = (G, σ1 ) and Γ2 = (G, σ2 ) gives a switching function θ : V → {+1, −1} and using which the matrix S, for the signature similarity, can be taken as S = diag(θ(v1 ), θ(v2 ), . . . , θ(vn )).

On signed paths, signed cycles and their energies

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We consider mainly the adjacency matrix and the Laplacian matrix (also called Kirchhoff matrix) which are direct generalization of the respective matrices for unsigned graphs. The adjacency matrix of the signed graph Γ = (G, σ) where G = (V, E) is A(Γ) = (aij ) is defined as  σ(vi vj ) if vi vj ∈ E(Γ) aij = 0 otherwise Laplacian matrix of a signed graph is the adjacency matrix in the signs reversed and with degrees of the vertices inserted in the diagonal. That is L(Γ) = D(Γ) − A(Γ). The ordinary adjacency and Laplacian matrices of a graph G are identical with those of the all-positive signed graph +G. The so-called signless Laplacian of an unsigned graph G is the Laplacian matrix of the allnegative graph −G. Moreover, for a signed graph Γ = (G, σ), the signless Laplacian is nothing but the Laplacian of −Γ = (G, −σ). As the adjective ’signless’ does not suit for signed graphs as the matrix contains positive and negative entries, we call it as Q matrix of a signed graph. Eigenvalues of the adjacency matrix, the Laplacian matrix and the Q matrix of a graph have been widely used to characterize properties of a graph and extract some useful information from its structure. The eigenvalues of the adjacency matrix of a graph are often referred to as the eigenvalues of the graph and those of the Laplacian matrix as the Laplacian eigenvalues.

1. Main Results In this section, we investigate the eigenvalues of the adjacency, Laplacian and Q matrices of signed paths, and using which, their energies are computed. We also show that all signed paths of size n are equienergetic under the respective matrices. Moreover, recurrence relations for the characteristic polynomials of signed paths in the case of Laplacian and Q matrices are established. (r) We denote by Pn , where 0 ≤ r ≤ n − 1, signed paths of order n and size n − 1 with r negative edges where the underlying graph is the path Pn . Note (0) (n−1) (r) that Pn = Pn and Pn = −Pn . The characteristic polynomials of A(Pn ), (r) (r) (r) (r) L(Pn ) and Q(Pn ) for 0 ≤ r ≤ n − 1, are denoted by Φ(Pn ; λ), Ψ(Pn ; λ) (r) and η(Pn ; λ) respectively. For brevity, they are also, respectively, denoted by Φn (λ) , Ψn (λ) and ηn (λ). The eigenvalues of the matrix M of order n are denoted λj (M) for j = 1, 2, . . . , n and at times we drop the letter M if no confusion arises. The energy E(Γ) of a signed graph Γ is the sum of the absolute values of the eigenvalues of its adjacency matrix. The Laplacian energy of Γ, denoted

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Germina K.A. and Shahul Hameed K

by EL (Γ), is defined as EL (Γ) =

|V | 

¯ |λj (L(Γ)) − d(Γ)|,

j=1

¯ = 2|E|/|V | is the average where L(Γ) is the Laplacian matrix of Γ and d(Γ) degree of the vertices in Γ. These definitions are direct generalizations of those used for unsigned graphs ([5] for energy and [6] for Laplacian energy). We take the Q-energy,EQ (Γ), as EQ (Γ) =

|V | 

¯ |λj (Q(Γ)) − d(Γ)|

j=1

Following two theorems due to B.D Acharya and M.K Gill are the sources of inspiration for this paper. Theorem 1.1. ([4]) If Γ = (G, σ) is a signed graph and v is any vertex of Γ and w is a vertex adjacent to v, then Φ(Γ; λ) satisfies the recurrence relation Φ(Γ; λ) = λΦ(Γ − v; λ) −

 w∼v

(1.1)





Φ(Γ − v − w; λ) − 2(



Φ(Γ − V (Z); λ)

Z∈C + (v)

Φ(Γ − (V (Z)); λ))

Z∈C − (v)

where C + (v) and C − (v), respectively, are the set of positive and negative cycles containing v. Theorem 1.2. ([1]) if and only if (1.2)

If Γ = (G, σ) is a signed graph, then Γ is cycle balanced Φ(Γ; λ) = Φ(G; λ)

Y. Hou, J. Lie and Y. Pan[11] proved the following result which provides another spectral criterion for cycle balance in signed graphs using the Laplacian matrix. Theorem 1.3. ([11]) A connected signed graph Γ = (G, σ) is cycle balanced if and only if det(L(Γ)) = 0. From the definition of the switching similar matrices and switching equivalence of two signed graphs, it is given in [11] that : Theorem 1.4. ([11]) Two signed graphs, Γ1 = (G, σ1 ) and Γ2 = (G, σ2 ) on the same underlying graph G, are switching equivalent if and only if their Laplacian matrices are signature similar.

On signed paths, signed cycles and their energies (r)

(r1 )

Observing the facts that Pn contains no cycles and Pn the following two results.

3459 (r )

∼ Pn 2 , we have

(r)

Corollary 1.5. Each Pn , for 0 ≤ r ≤ n − 1, has one of its Laplacian eigenvalues as zero. (r)

Corollary 1.6. Pn , where 0 ≤ r ≤ n − 1, have both its adjacency and Laplacian eigenvalues as that in the respective cases of Pn . Theorem 1.7. ([3]) The second order difference equation xn = axn−1 +bxn−2 , where a and b are fixed constants, b = 0, has the solution of the form  a + √a2 + 4b n  a − √a2 + 4b n (1.3) +β f or n = 0, 1, 2 . . . xn = α 2 2 Invoking Theorem 1.2 of Acharya[1], we have Lemma 1.8. ( Recurrence formula for the characteristic polynomial of signed paths) (r) Φ(Pn ; λ) satisfies the recurrence relation (r)

(r)

(1.4)

Φ(Pn(r) ; λ) = λΦ(Pn−1 ; λ) − Φ(Pn−2 ; λ) f or 0 ≤ r ≤ n − 1.

Proof.

The proof follows from the fact that Pn

(r)

does not contain any cycles.

(r)

Eigenvalues of A(Pn ) for r = 0 is given in [2]. An alternate proof for the (r) signed paths Pn , where 0 ≤ r ≤ n − 1, using difference equation is given in Theorem 1.9. (r)

The eigenvalues of A(Pn ) are given by  πj  (1.5) f orj = 1, 2 . . . , n λj = 2cos n+1 Proof. We have from equation 1.4, Φn (λ) = λΦn−1 (λ) − Φn−2 (λ). This is a difference equation of the form xn = axn−1 + bxn−2 whose solution is given in equation 1.3. Therefore, we have Theorem 1.9.

(1.6)

√ λ − λ2 − 4 n λ2 − 4 n ) + β( ) where n = 0, 1, 2, . . . = α( Φn (λ) = 2 2 To evaluate α and β we use the convention that Φ0 (λ) = Φ(P0 ; λ) = 1 and the these initial conditions in equation fact that Φ1 (λ) = Φ(P1 ; λ) = λ. Applying √ λ+√ λ2 −4 1.6, we get β = 1 − α and α = 2 λ2 −4 Thus we have, Φ(Pn(r) ; λ)

(1.7) Φ(Pn(r) ; λ)

=√

1 λ2 − 4

λ+



 λ + √λ2 − 4 n+1 2



 λ − √λ2 − 4 n+1  2

where n = 0, 1, 2, . . .

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Taking, λ = 2cosθ, we get from equation 1.7, 1  2cosθ + 2isinθ n+1  2cosθ − 2isinθ n+1  − Φn (2cosθ) = 2isinθ 2 2 =

(−icosecθ) i(n+1)θ (e − e−i(n+1)θ ) 2

i.e., (1.8)

Φn (2cosθ) = cosecθ sin (n + 1)θ

πj , for j = 1, 2, . . . , n. Then the Solving we get (n + 1) θ = π j or θ = n+1 πj eigenvalues are given by λj = 2cos θ = 2cos n+1 , for j = 1, 2, . . . , n.

Corollary 1.10. E(Pn(r) ) = 2

(r)

The energy, E(Pn ), is given by: n 

|cos(

j=1

πj )|, for all r such that 0 ≤ r ≤ n − 1. n+1

(r)

Hence Pn , for all r such that 0 ≤ r ≤ n − 1, are equienergetic under the adjacency matrix. Corollary 1.11.

(r)

(r)

The energy, E(Pn ), satisfies E(Pn ) ≤ 2n. (r)

Proof. The fact that |cosθ| ≤ 1 in the formula for E(Pn ) in the Corollary 1.10 implies the required inequality. Theorem 1.12. (Recurrence formula for the characteristic poly(r) nomial of the Laplacian of signed paths) Ψ(Pn ; λ) satisfies the recurrence relation (1.9)

(r)

(r)

(λ − 2)Ψ(Pn−1 ; λ) =  λ − 1 1 0   1 λ − 2 1   0 1 λ−2   0 0 1 (λ − 2)  . . . . . . . ..   ... ... ...   0 0 ...   0 0 ... Proof.

(r)

Ψ(Pn(r) ; λ) = (λ − 2)Ψ(Pn−1 ; λ) − Ψ(Pn−2 ; λ)

0 0 1 λ−2 ... ... ... ...

0 0 0 1 ... ... ... ...

 ... 0 0   ... 0 0  ... 0 0  ... 0 0  ... ... . . .  ... ... . . .  ... λ −2 1  ... 1 λ − 1(n−1)×(n−1)

On signed paths, signed cycles and their energies

3461

   1 0 0 0 0 ... 0 0      1 λ−1 1 0 0 . . . 0 0     0 1 λ − 2 1 0 . . . 0 0    0 0 1 λ − 2 1 ... 0 0   = (λ − 2)  ... ... ... ... ... . . .  . . . . . . . . . . . . ... ... ... ... ... . . .    0 0 ... ... ... ... λ −2 1    0 0 ... ... ... ... 1 λ − 1n×n Now applying row and column operations, it is seen that the expression on the left hand side    0 −1 1 0 0 ... 0 0      1 λ−2 1 0 0 . . . 0 0    0 1 λ−2 1 0 ... 0 0    0 0 1 λ −2 1 ... 0 0  (r) = Ψ(Pn ; λ) +  ... ... ... ... ... . . .  . . . . . . . . . . . . ... ... ... ... ... . . .    0 0 ... ... ... ... λ −2 1    0 0 ... ... ... ... 1 λ − 1n×n   −1 1 0 0 0 ... 0 0      1 λ−2 1 0 0 . . . 0 0     0 1 λ − 2 1 0 . . . 0 0    0 0 1 λ − 2 1 ... 0 0  (r)  = Ψ(Pn ; λ) −  ... ... ... ... ... . . .  . . . . . . . . . . . . ... ... ... ... ... . . .    0 0 ... ... ... ... λ − 2 1    0 0 ... ... ... ... 1 λ − 1(n−1)×(n−1)   λ − 1 1 0 0 0 ... 0 0     1  λ − 2 1 0 0 . . . 0 0    0 1 λ−2 1 0 ... 0 0    0 0 1 λ − 2 1 ... 0 0  (r) = Ψ(Pn ; λ)+ ... ... ... ... ... ... . . .   ...  ... ... ... ... ... ... ... . . .    0 0 ... ... ... ... λ −2 1    0 0 ... ... ... ... 1 λ − 1(n−2)×(n−2) (r)

(r)

= Ψ(Pn ; λ) + Ψ(Pn−2 ; λ). Theorem 1.13.

(1.10)

(r)

The non-zero Laplacian eigenvalues of Pn are given by

λj = 2(1 + cos

πj ) f or j = 1, 2, . . . , n − 1. n

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Germina K.A. and Shahul Hameed K

Also each λj is distinct and that λj > 0, f or j = 1, 2, . . . , n − 1.

(1.11)

Proof. Proceeding as in Theorem 1.9 , adopting the convention thatΨ(P0 ; λ) = (r) 0 and using the fact thatΨ(P1 ; λ) = λ we get, Ψ(Pn(r) ; λ)

 (λ − 2) + (λ − 2)2 − 4 n  (λ − 2) − (λ − 2)2 − 4 n  = − 2 2 (λ − 2)2 − 4 λ

where n = 0, 1, 2, . . . Put λ − 2 = 2cos θ. Then, Ψn (2 + 2cosθ) =

(2 + 2cosθ)  2cosθ + 2isinθ n  2cosθ − 2isinθ n  − 2isinθ 2 2  1 + cosθ  inθ (e − e−inθ ) = −i sinθ

That is, (1.12)

Ψn (2 + 2cosθ) = 2

sin nθ (1 + cosθ) sinθ

Solving for non-zero eigenvalues, we get n θ = π j or θ = πj , for j = n πj 1, 2, . . . , n − 1. Then λj = 2 + 2cos θ = 2(1 + cos n ), for j = 1, 2, . . . , n − 1. πj ) = 2sin2 ( 2n ). Inequality 1.11 follows from the fact 1 + cos( πj n Theorem 1.14.

(r)

The Laplacian energy, EL (Pn ), is given by

(1.13) EL (Pn(r) )

=2

n  j=1

|cos(

1 πj ) + | f or all integer r such that 0 ≤ r ≤ n − 1. n n

(r)

Hence, Pn , for all r such that 0 ≤ r ≤ n − 1, are equienergetic under the Laplacian matrix. That is, they are Laplacian equienergetic. Corollary 1.15. (1.14)

(r)

The Laplacian energy, EL (Pn ), satisfies EL (Pn(r) ) ≤ 2(n + 1). (r)

Remark 1.16. From the Laplacian eigenvalues of signed paths Pn given by equation 1.10 it is clear that |λj | ≤ 4, so there are only five integral eigenvalues 0, 1, 2, 3 and 4. The eigenvalue 4 is never attained by any signed paths. (r)

Corollary 1.17. The only signed paths Pn , which have the Laplacian eigenvalues 1 and 3 are given for n = 3, 6, 9, . . . and signed paths which have the Laplacian eigenvalue 2 are, for n = 2, 4, 6, . . .

On signed paths, signed cycles and their energies (r)

3463

(r)

Now we deal with the Q matrix Q(Pn ) of the path Pn , where 0 ≤ r ≤ n − 1 and its energy. (r)

(r)

Lemma 1.18. For each Pn , where 0 ≤ r ≤ n − 1, Q(Pn ) is signature similar to Q(Pn ). (r)

(r)

Proof. As in the case of L(Pn ), since each Pn is switching equivalent to the underlying path Pn , we get a diagonal (−1, 1)- matrix S, which is invertible, associated with the switching function. For example, if the edge ei = vj vk is negative, then the diagonal elements in S corresponding to the vertices vj and vk , respectively, be chosen either as +1 and -1 or as -1 and +1. If the edge is positive, then the selection of the diagonal element can be done either as -1 and -1 or as +1 and +1, respectively. Thus, SQ(Pn(r) )S −1 = S(D(Pn(r)) + A(Pn(r) ))S −1 = SD(Pn(r) )S −1 + SA(Pn(r) )S −1 = D(Pn ) + A(Pn ) = Q(Pn ) (r)

showing that Q(Pn ) is signature similar to Q(Pn ). Theorem 1.19. (Recurrence formula for the characteristic poly(r) nomial of Q(Pn )) (r) η(Pn ; λ) satisfies the recurrence relation (r)

(r)

η(Pn(r) ; λ)) = (λ − 2)η(Pn−1 ; λ) − η(Pn−2 ; λ)

(1.15)

Proof. The result can be obtained as in Theorem 1.12 and using the fact ob(r) served in the Lemma 1.18 that Q(Pn ) is signature similar to Q(Pn ). Observe (r ) (r ) also that Q(Pn i )=L(Pn j ) for some ri and rj such that 0 ≤ ri , rj ≤ n − 1. (r)

Corollary 1.20. (1.16)

The non-zero Q-eigenvalues of Pn are given by λj = 2(1 + cos

πj ) f or j = 1, 2, . . . , n − 1. n (r)

(r)

Proof. The result follows since η(Pn ; λ) and Ψ(Pn ; λ) satisfy the same recurrence relation. Theorem 1.21.

(r)

1. The Q-energy, EQ (Pn ), is given by

(1.17) EQ (Pn(r) ) = 2

n  j=1 (r)

|cos(

πj 1 ) + |f or all integer r such that 0 ≤ r ≤ n − 1. n n

Hence, Pn , for all r such that 0 ≤ r ≤ n − 1, are equienergetic under the matrix Q.

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Germina K.A. and Shahul Hameed K (r)

2. The Q-energy, EQ (Pn ), satisfies: EQ (Pn(r) ) ≤ 2(n + 1).

(1.18)

(r)

3. The only signed pathsPn , which have the Q-eigenvalues 1 and 3,are for (r) n = 3, 6, 9, . . . and the only signed paths Pn , which have the Q-eigenvalue 2,are, for n = 2, 4, 6, . . . 1.1. Signed cycles and their energy. Before we conclude, it is worthwhile (r) to have a look at the relation between signed cycles Cn with r negative edges (r) for 0 ≤ r ≤ n and signed paths Pn with r negative edges for 0 ≤ r ≤ n − 1. Many of the formulas that follow have cases depending on the parity of a parameter. Therefore, for an integer r, we define [r] = 0 if r is even, (r) [r] = 1 if r is odd. A.M. Mathai [9] has given the eigenvalues of Cn with odd and even number of negative edges as (r)

Theorem 1.22. ([9]) Eigenvalues λj of Cn for j = 1, 2 . . . n are given by ). λj = 2cos( (2j−[r])π n Theorem 1.22 has the following important consequences. Corollary 1.23. (r)

1. The energy, E(Cn ), for 0 ≤ r ≤ n is given by: E(Cn(r) )

=2

n 

|cos(

j=1

(2j − [r])π )| n (r)

2. Laplacian eigenvalues and the corresponding energies of Cn by: λj (L(Cn(r) )) = 2 − 2cos( EL (Cn(r) ) = 2

n 

|cos(

j=1

are given

(2j − [r])π ) n

(2j − [r])π )| n (r)

3. Q-eigenvalues and the corresponding Q-energies of Cn are given by: λj (Q(Cn(r) )) = 2 + 2cos( EQ (Cn(r) )

=2

n  j=1

|cos(

(2j − [r])π ) n

(2j − [r])π )| n

On signed paths, signed cycles and their energies

3465

Proof. 1. The proof follows from the definition of the energy and Theorem 1.22. (r) 2. Here we observe that Cn is a regular signed graph with degree of each (r) (r) vertex being 2. Hence L(Cn ) = 2I − A(Cn ), where I is the identity matrix of order n. So the eigenvalues of the Laplacian matrix are given by (r) subtracting the eigenvalues of A(Cn ) from 2. The remaining is obvious. (r) (r) 3. Here we have Q(Cn ) = 2I + A(Cn ). Therefore, Q-eigenvalues are given (r) by adding 2 to the eigenvalues of A(Cn ). Corollary 1.24. E(Cn(r) ) = EL (Cn(r) ) = EQ (Cn(r) ). (r)

The following Corollary, which connects the characteristic polynomials of Cn (r) and Pn , is again obtained by using equation 1.1. Corollary 1.25. (r )

3 Φ(Cn(r1 ) ; λ) = Φ(Pn(r2 ) ; λ) − Φ(Pn−2 ; λ) ± 2

for all r1 , r2 and r3 such that 0 ≤ r1 ≤ n,0 ≤ r2 ≤ n − 1 and 0 ≤ r3 ≤ n − 2 ac(r ) cording as the deleted edge from Cn 1 is negative or positive. (r)

Proof. Using equation 1.1 and adopting the convention that Φ(C0 ; λ) = 1, we get the required result. 2. Scope and Conclusion We have identified the following problems for further investigation. Problem 1: Find a recurrence formula for the characteristic polynomial of the Laplacian matrix of a signed graph. Signed cycles are found to have same energy under the three matrices considered here. So we propose Problem 2: Find or characterize all signed graphs which have the same energy for their adjacency, Laplacian and the Q matrices. Acknowledgements The second author is indebted to the University Grants Commission(UGC) for granting him Teacher Fellowship under UGC’s Faculty Development Programme during XI plan. References [1 ] B D Acharya, Spectral Criterion for Cycle Balance in Networks.Journal of Graph Theory, Vol.4(1980) 1-11.

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Germina K.A. and Shahul Hameed K

[2 ] D.M.Cvetkovic,M.Doob,H. Sachs, Spectra of Graphs, third ed., Johann Abrosius Barth Verlag, 1995 (First edition:DeutscherVerlag derWissenschaften, Berlin 1980; AcademicPress, NewYork 1980). [3 ] Fuzhen Zhang, Matrix Theory: Basic Theory and Techniques. SpringerVerlag (1999). [4 ] M K Gill and B D Acharya , A Recurrence Formula for Computing the Characteristic Polynomial of a Sigraph. Journal of Combinatorics, Vol.1(1980) 68-72. [5 ] I. Gutman, The energy of a graph, Ber.Math.-Stat. Sekt. Forschungszent. Graz 103 (1978) 122. [6 ] I. Gutman and Bo Zhou, Laplacian energy of a graph, Linear Algebra Appl.414 (1978) 2937. [7 ] F. Harary, R.Z. Norman, and D. Cartwright: Structural Models: An Introduction to the Theory of Directed Graphs. Wiley, New York (1965). [8 ] F. Harary, Graph Theory. Addison Wesley, Reading, MA (1972). [9 ] A.M. Mathai,On Adjacency Matrices of Simple Signed cyclic connected graphs.Submitted(2010). [10 ] F.S. Roberts, Graph Theory and Its Application to Problems of Society, SIAM, Philadelphia, Pennsylvania, 1978. [11 ] Yaoping Hou, Jiongsheng Li, and Yongliang Pan, On the Laplacian Eigenvalues of Signed Graphs. Linear and Multilinear Algebra Vol.51(2003) 21-30. [12 ] T. Zaslavsky, Matrices in the theory of signed graphs.In: International Conference on Discrete Mathematics (ICDM 2008) and Graph Theory Day IV (Proc. [Lecture Notes], Mysore, (2008), pp. 187-198. [13 ] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas. VII Edition, Electronic J. Combinatorics, 8(1998), Dynamic Surveys, #8, 124 pp. Received: July, 2010