On Net-Regular Signed Graphs

Transcription

1 Iteratioal J.Math. Combi. Vol.1(2016), O Net-Regular Siged Graphs Nuta G.Nayak Departmet of Mathematics ad Statistics S. S. Dempo College of Commerce ad Ecoomics, Goa, Idia ayakuta@yahoo.com Abstract: A siged graph is a ordered pair Σ = (G, σ), where G = (V, E) is the uderlyig graph of Σ ad σ : E {+1, 1}, called sigig fuctio from the edge set E(G) of G ito the set {+1, 1}. It is said to be homogeeous if its edges are all positive or egative otherwise it is heterogeeous. Siged graph is balaced if all of its cycles are balaced otherwise ubalaced. It is said to be et-regular of degree k if all its vertices have same et-degree k i.e. k = d ± Σ (v) = d+ Σ (v) d Σ (v), where d+ Σ (v)(d Σ (v)) is the umber of positive(egative) edges icidet with a vertex v. I this paper, we obtaied the characterizatio of et-regular siged graphs ad also established the spectrum for oe class of heterogeeous ubalaced et-regular siged complete graphs. Key Words: Smaradachely k-siged graph, et-regular siged graph,co-regular siged graphs, siged complete graphs. AMS(2010): 05C22, 05C Itroductio We cosider graph G is a simple udirected graph without loops ad multiple edges with vertices ad m edges. A Smaradachely k-siged graph is defied as a ordered pair Σ = (G, σ), where G = (V, E) is a uderlyig graph of Σ ad σ : E {e 1, e 2, e 3,, e k } is a fuctio, where e i {+, }. A Smaradachely 2-siged graph is kow as siged graph. It is said to be homogeeous if its edges are all positive or egative otherwise it is heterogeeous. We deote positive ad egative homogeeous siged graphs as +G ad G respectively. The adjacecy matrix of a siged graph is the square matrix A(Σ) = (a ij ) where (i, j) etry is +1 if σ(v i v j ) = +1 ad 1 if σ(v i v j ) = 1, 0 otherwise. The characteristic polyomial of the siged graph Σ is defied as Φ(Σ : λ) = det(λi A(Σ)), where I is a idetity matrix of order. The roots of the characteristic equatio Φ(Σ : λ) = 0, deoted by λ 1, λ 2,, λ are called the eigevalues of siged graph Σ. If the distict eigevalues of A(Σ) are λ 1 λ 2 λ ad their multiplicities are m 1, m 2,..., m, the the spectrum of Σ is Sp(Σ) = {λ (m1) 1, λ (m2) 2,, λ (m) }. Two siged graphs are cospectral if they have the same spectrum. The spectral criterio 1 Received May 19, 2015, Accepted February 15, 2016.

2 58 Nuta G.Nayak for balace i siged graph is give by B.D.Acharya as follows: Theorem 1.1([1]) A siged graph is balaced if ad oly if it is cospectral with the uderlyig graph. i.e. Sp(Σ) = Sp(G). The sig of a cycle i a siged graph is the product of the sigs of its edges. Thus a cycle is positive if ad oly if it cotais a eve umber of egative edges. A siged graph is said to be balaced (or cycle balaced) if all of its cycles are positive otherwise ubalaced. The egatio of a siged graph Σ = (G, σ), deoted by η(σ) = (G, σ) is the same graph with all sigs reversed. The adjacecy matrices are related by A( Σ) = A(Σ). Theorem 1.2([12]) Two siged graphs Σ 1 = (G, σ 1 ) ad Σ 2 = (G, σ 2 ) o the same uderlyig graph are switchig equivalet if ad oly if they are cycle isomorphic. I siged graph Σ, the degree of a vertex v is defied as sdeg(v) = d(v) = d + Σ (v) + d Σ (v), where d + Σ (v)(d Σ (v)) is the umber of positive(egative) edges icidet with v. The et degree of a vertex v of a siged graph Σ is d ± Σ (v) = d+ Σ (v) d Σ (v). It is said to be et-regular of degree k if all its vertices have same et-degree equal to k. Hece et-regularity of a siged graph ca be either positive, egative or zero. We deote et-regular siged graphs as Σ k. We kow [13] that if Σ is a k et-regular siged graph, the k is a eigevalue of Σ with j as a eigevector with all 1 s. K.S.Hameed ad K.A.Germia [6] defied co-regularity pair of siged graphs as follows: Defiitio 1.3([6]) A siged graph Σ = (G, σ) is said to be co-regular, if the uderlyig graph G is regular for some positive iteger r ad Σ is et-regular with et-degree k for some iteger k ad the co-regularity pair is a ordered pair of (r, k). The followig results give the spectra of siged paths ad siged cycles respectively. Lemma 1.4([3]) The siged paths P (r), where r is the umber of egative edges ad 0 r 1, have the eigevalues(idepedet of r) give by λ j = 2 cos πj, j = 1, 2,,. + 1 Lemma 1.5([9]) The eigevalues λ j of siged cycles C (r) λ j = 2 cos (2j [r])π, j = 1, 2,, ad 0 r are give by where r is the umber of egative edges ad [r] = 0 if r is eve, [r] = 1 if r is odd. Spectra of graphs is well documeted i [2] ad siged graphs is discussed i [3, 4, 5, 9]. For stadard termiology ad otatios i graph theory we follow D.B.West [10] ad for siged graphs T. Zaslavsky [14]. The mai aim of this paper is to characterize et-regular siged graphs ad also to prove

3 O Net-Regular Siged Graphs 59 that there exists a et-regular siged graph o every regular graph but the coverse does ot hold good. Further, we costruct a family of coected et-regular siged graphs whose uderlyig graphs are ot regular. We established the spectrum for oe class of heterogeeous ubalaced et-regular siged complete graphs. 2. Mai Results Spectral properties of regular graphs are well kow i graph theory. Theorem 2.1([2]) If G is a r regular graph, the its maximum adjacecy eigevalue is equal to r ad r = 2m. Here we geeralize Theorem 2.1 to siged graphs as graph is cosidered as oe case i siged graph theory. We deote total umber of positive ad egative edges of Σ as m + ad m respectively. The followig lemma gives the structural characterizatio of siged graph Σ so that Σ is et-regular. Lemma 2.2 If Σ = (G, σ) is a coected et-regular siged graph with et degree k the k = 2M, where M = (m+ m ), m + is the total umber of positive edges ad m is the total umber of egative edges i Σ. Proof Let Σ = (G, σ) be a et-regular siged graph with et degree k. The by defiitio, d ± Σ (v) = d+ Σ (v) d Σ (v). Hece, d ± Σ (v) = d + Σ (v) d Σ (v). i=1 i=1 i=1 Thus, Whece, k = d + Σ (v) d Σ (v). i=1 i=1 [ k = 1 ] d + Σ (v) d Σ (v) = 1 [ 2m + 2m ] i=1 i=1 [ 2(m + m ] ) = = 2M. ¾ Corollary 2.3 If Σ = (G, σ) is a siged graph with co-regularity pair (r, k) the r k. Proof Let Σ be a k et-regular siged graph the by Lemma 2.2, k = 2M, where M = (m + m ). Sice G is its uderlyig graph with regularity r o vertices the r = 2m,where m = m + + m. It is clear that 2m 2(m+ m ). Hece r k. ¾ Remark 2.4 By Corollary 2.3, if Σ = (G, σ) is a siged graph with co-regularity pair (r, k) o

4 60 Nuta G.Nayak vertices the r k r. Now the questio arises whether all regular graphs ca be et-regular ad vice-versa. From Lemma 2.2, it is evidet that at least two et-regular siged graphs exist o every regular graph whe m + = 0 or m = 0. We feel the coverse also holds good. But cotrary to the ituitio, the aswer is egative. Next result proves that uderlyig graph of all et-regular siged graphs eed ot be regular. Theorem 2.5 Let Σ be a et-regular siged graph the its uderlyig graph is ot ecessarily a regular graph. Proof Let ] Σ be a et-regular siged graph with et degree k. The by Lemma 2.2, k =. By chagig egative edges ito positive edges we get k = 2m where [ 2(m + m ) m = m + + m. If k = 2m is a positive iteger the uderlyig graph is of order k = r. If k = 2m is ot a positive iteger the k r. Hece the uderlyig graph of a et-regular siged graph eed ot be a regular graph. ¾ Shahul Hameed et.al. [7] gave a example of a coected siged graph o = 5 whose uderlyig is ot a regular graph. Here we costruct a ifiite family of et-regular siged graphs whose uderlyig graphs are ot regular. Example 2.6 Here is a ifiite family of et-regular siged graphs with the property that whose uderlyig graphs are ot regular. Take two copies of C, joi at oe vertex ad assig positive ad egative sigs so that degree of the vertex commo to both cycles will have et degree 0 ad also assig positive ad egative sigs to other edges i order to get et-degree 0. The resultat siged graph is a et-regular siged graph with et-degree 0 whose uderlyig graph is ot regular. We deote it as Σ (0) (2 1) for each C ad illustratio is show i Fig.1, 2 ad 3. I chemistry, uderlyig graphs of these siged graphs are kow as spiro compouds. I the followig figures, solid lies represet positive edges ad dotted lies represet egative edges respectively. Fig.1 Net-regular siged graph Σ 0 5 for C 3

5 O Net-Regular Siged Graphs 61 Fig.2 Net-regular siged graph Σ 0 7 for C 4 Fig.3 Net-regular siged graph Σ 0 9 for C 5 From Figures 1, 2 ad 3, we ca see that Σ 0 7 is a bipartite siged graph, but Σ0 5 ad Σ0 9 are o-bipartite siged graphs. The spectrum of these et -regular siged graphs are Sp(Σ 0 5)= {±2.2361, ±1, 0}, Sp(Σ 0 7 )= {±2.4495, ±1.4142, (0)3 }, Sp(Σ 0 9)={±2.3028, ±1.6180, ±1.3028, ±0.6180, 0}. Remark 2.7 Spectrum of this family of coected siged graphs Σ (0) (2 1) satisfy the pairig property i.e. spectrum is symmetric about the origi ad also these are o-bipartite whe cycle C is odd. Heterogeeous siged complete graphs which are cycle isomorphic to the uderlyig graph +K will have the spectrum {( 1), ( 1) ( 1) } ad which are cycle isomorphic to K will have the spectrum {(1 ), (1) ( 1) }. Here we established the spectrum for oe class of heterogeeous ubalaced et-regular siged complete graphs. Let C be a cycle o vertices ad C be its complemet where 4. Defie σ :

6 62 Nuta G.Nayak E(K ) {1, 1} by 1, if e C σ(e) = 1, if e C The Σ = (K, σ) is a ubalaced et-regular siged complete graph ad we deote it as K et where 4. The followig spectrum for K et is give by the author i [10]. Lemma 2.8([10]) Let K et the Sp(K et ) = be a heterogeeous ubalaced et-regular siged complete graph cos(2πj ) 1 1 where (5 ) gives the et-regularity of K et. : j = 1,..., 1, Lemma 2.9 ω r + ω r = 2 cos 2rπj uity. for 1 j ad 1 r, where ω is the th root of Proof Let 1 j ad 1 r. ω r + ω r = e 2rπij = cos 2rπj = 2 cos 2rπj. + e 2rπij e 2rπij 2rπj + i si = e 2rπij + cos 2rπj + e 2rπij i si 2rπj ¾ By usig the properties of the permutatio matrix [8] ad from Lemma 2.9, we give a ew spectrum for K et. Theorem 2.10 Let K et odd the ad if is eve the be a heterogeeous siged complete graph as defied above. If is 1 Sp(K et ) = {2 cos 2πj 2 r=2 2 cos 2rπj : 1 j } 2 Sp(K et ) = {2 cos 2πj 2 cosπj 2 cos 2rπj r=2 : 1 j }. Proof Label the vertices of a circulat graph as 0, 1,, ( 1). The the adjacecy

7 O Net-Regular Siged Graphs 63 matrix A is A = 0 c 1 c 2 c 1 c 1 0 c 1 c 2 c 2 c 1 0 c 3.. c 1 c 2 c 3 0, where c i = c i = 0 if vertices i ad i are ot adjacet ad c i = c i = 1 if vertices i ad i are adjacet. Hece A = = c 1 P 1 + c 2 P c 1 P 1 1 c r P r, r=1 where P is a permutatio matrix. Let K et be the heterogeeous siged complete graph ad A(K et ) be its adjacecy matrix. A(K et) is a circulat matrix with first row [0, 1, 1, 1,, 1, 1]. Here c 1 = 1, c 2 = 1, c 3 = 1,, c 1 = 1. Hece A(K et ) ca be writte as a liear combiatio of permutatio matrix P. A(K et) = P 1 P 2 P 3 P 2 + P 1. Case 1. If is odd the } A(K et ) = {(P 1 + P 1 ) (P 2 + P 2 ) (P P +1 2 ) ad ω Sp(P). Hece Sp(K et ) = = Sp(K et ) = {(ω 1 + ω 1 ) (ω 2 + ω 2 ) (ω ω +1 2 ) } { 2 cos 2πj 1 2( 2 2 cos )πj } 1 2πj 2 2 cos 2 cos 2rπj : 1 j r=2 Case 2. If is eve the { } A(K et ) = (P 1 + P 1 ) (P 2 + P 2 ) (P P +1 2 ) (P 2 ) ad ω Sp(P). Hece { } Sp(K et ) = (ω 1 + ω 1 ) (ω 2 + ω 2 ) (ω ω +1 2 ) (ω 2 ) { = 2 cos 2πj 2 cos 2( 1 2 )πj cos 2( 2 )πj } : 1 j.

8 64 Nuta G.Nayak So Sp(K et ) = 2πj 2 cos cosπj 2 2 r=2 2 cos 2rπj : 1 j. ¾ Ackowledgemet The author thaks the Uiversity Grats Commissio(Idia) for providig grats uder mior research project No /14 durig XII pla. Refereces [1] B.D.Acharya, Spectral criterio for cycle balace i etworks, J. Graph Theory, 4(1980), [2] D.M.Cvetkovic, M.Doob, H.Sachs, Spectra of Graphs, Academic Press, New York, [3] K.A.Germia, K.S.Hameed, O siged paths, siged cycles ad their eergies, Applied Math Sci., 70(2010) [4] K.A.Germia, K.S.Hameed, T.Zaslavsky, O product ad lie graphs of siged graphs, their eigevalues ad eergy, Liear Algebra Appl., 435(2011) [5] M.K.Gill, B.D.Acharya, A recurrece formula for computig the characteristic polyomial of a sigraph, J. Combi. Iform. Syst. Sci., 5(1)(1980) [6] K.S.Hameed, K.A.Germia, O compositio of siged graphs, Discussioes Mathematicae Graph Theory, 32(2012) [7] K.S.Hameed, V.Paul, K.A.Germia, O co-regular siged graphs, Australasia Joural of Combiatorics, 62(2015) [8] R.A.Hor, C.R.Johso, Matrix Aalysis, Cambridge Uiversity Press, Cambridge, [9] A.M.Mathai, T.Zaslavsky, O Adjacecy matrices of simple siged cyclic coected graphs, J. of Combiatorics, Iformatio ad System Scieces, 37(2012) [10] N.G.Nayak, Equieergetic et-regular siged graphs, Iteratioal Joural of Cotemporary Mathematical Scieces, 9(2014) [11] D.B.West, Itroductio to Graph Theory, Pretice-Hall of Idia Pvt. Ltd., [12] T.Zaslavsky, Siged graphs, Discrete Appl.Math., 4(1982) [13] T.Zaslavsky, Matrices i the theory of siged simple graphs, Advaces i Discrete Mathematics ad Applicatios, (Ramauja Math. Soc. Lect. Notes Mysore, Idia), 13(2010) [14] T.Zaslavsky, A mathematical bibliography of siged ad gai graphs ad allied areas, (Mauscript prepared with Marge Pratt), Joural of Combiatorics, DS, NO.8(2012), pp