Keywords: complete graph, coloursignlesslaplacian matrix, coloursignlesslaplacian energy of a graph.

Transcription

1 Amerca Iteratoal Joural of Research Scece, Techology, Egeerg & Mathematcs Avalable ole at ISSN (Prt): , ISSN (Ole): , ISSN (CD-ROM): AIJRSTEM s a refereed, dexed, peer-revewed, multdscplary ad ope access joural publshed by Iteratoal Assocato of Scetfc Iovato ad Research (IASIR), USA (A Assocato Ufyg the Sceces, Egeerg, ad Appled Research) RELATION BETWEEN COLOUR LAPLACIAN ENERGY AND COLOUR SIGNLESS LAPLACIAN ENERGY OF A COMPLETE GRAPH K. Ameeal Bb 1, B. Vjayalakshm, M. Malath 3 Departmet of Mathematcs D.K.M College for Wome (Autoomous), Vellore, Tamladu, Ida Abstract: Let G be a smple, fte, coected ad udrected graph of order ad sze m. Let λ 1, λ,...,λ be the egevalues of the colour adjacecy matrx of G, ad let µ 1, µ,...,µ be the egevalues of the colour Laplaca matrx. The, the coloursglesslaplaca eergy of G s defed as LE + C (G) = m, where m s the average degree of the graph G. I ths paper, we foud the relato betwee colourlaplaca eergy ad coloursglesslaplaca eergy of complete graph ad also attaed ther bouds. Keywords: complete graph, coloursglesslaplaca matrx, coloursglesslaplaca eergy of a graph. I. Itroducto Let G be a smple, fte, coected ad udrected graph wth vertex set V(G)={v 1,v,.v } ad edges E(G) = {e 1,e, e m}. Let A(G) ad D(G) be the adjacecy matrx ad the dagoal matrx wth the vertex degrees of the graph G o the dagoal respectvely. The matrces L(G) = D(G) A(G) ad L + (G) = D(G) + A(G) are the sglesslaplaca matrces of the graph G. For more results o the spectral propertes of sglesslaplaca matrx, oe may refer to [1],[],[3],[4],[5],[6],[7]. Let {µ 1,µ,...µ } be the sglessege values of the graph G. m The sglesslaplaca eergy [8] of the graph G s defed as LE + (G)=. A colourg of a graph G s a assgmet of colours to ts vertces such that o two adjacet vertces share the same colour. The mmum umber of colours eeded for the colourg of a graph G s called the chromatc umber of G ad s deoted by χ(g). Let G be a coloured graph. The etres of the colour adjacecy matrx A C(G) are as follows: If C(v ) s the colour of vertex v, the 1, f v ad v j are adjacet wth C(v ) C(v j ) a j = { 1, f v ad v j are o adjacet wth C(v ) = C(v j ) 0, otherwse The ege values {λ 1,λ,...λ } of A C(G) are called the colourege values of G. The colour eergy of a graph deoted by E C(G) s defed as the sum of the absolute values of the coloursglessege values of G,.e., E C(G)=. I 013,ch.Adga, E.Sampathkumar, M.A.Srraj ad A.S.shrkath [17],[18] have studed the eergy of the coloured graph G. II Prelmares A. Proper colourg of a graph A proper colourg of a graph s a assgmet of colours to the vertces of the graph so that o two adjacet vertces have the same colour. B.Complete graph A complete graph s a graph whch each par of graph vertces s coected by a edge. The assgmet of colours to the vertces of the complete graph are dfferet to each other. A complete graph k of vertces requres colours. So the chromatc umber of the complete graph k s. AIJRSTEM ; 018, AIJRSTEM All Rghts Reserved Page 08

2 Bb et al., Amerca Iteratoal Joural of Research Scece, Techology, Egeerg & Mathematcs, 3(1), Jue-August, 018, pp C. Laplaca matrx Let G be a graph wth order ad sze m. The Laplaca matrx of the graph G s deoted by L = (Lj) s a square matrx defed by L j= { 1, f v ad v j are adjacet 0, f v ad v j are o adjacet d, f v = v j Where d s the degree of the vertex v. D.Laplaca eergy Let G be a coected graph of order wth Laplaca egevalues μ 1 μ μ -1 μ = 0. The Laplaca s symmetrc postve semdefte. The Laplaca eergy of the graph m G s defed as LE(G) =. E.Sglesslaplaca matrx For a gve graph,the matrx Q = D + A s called the sglesslaplaca matrx, where A s the adjacecy matrx ad D s the dagoal matrx of vertex degrees d. F.Colourlaplaca eergy The colourlaplaca matrx of G s defed as L C(G) = D(G) - A C(G). The ege values {µ 1,µ,...µ } of L C(G) are called colourlaplacaege values of the graph G. m The colourlaplaca eergy of G deoted by LE C(G) =. G.Coloursglesslaplaca eergy Let G be a coloured graph o vertces ad m edges. We defed the coloursglesslaplaca matrx of G as L + C (G) = D(G) + A C(G). The ege values {µ + 1,µ +,...µ + } of L + C (G) are called coloursglesslaplacaege values of the graph G. Let G be a coloured graph of order ad sze m. The the coloursglesslaplaca eergy of G, deoted by m LE + C (G) =. H. Spectrum of G The set of graph ege values of the adjacecy matrx s called the spectrum of the graph. I.Colour spectrum of G The sum of the absolute values of the dstct colourege values of A C(G) areif μ 1 > μ > μ -1>μ r, r wth ther multplctes are m 1,m,...m r s defed as color spectrum of G s wrtte by Spec c(g)={ μ 1, μ,. μ r m 1, m,. m r }. J.Colourlaplaca spectrum of G If μ 1 > μ > μ -1>μ r, r are the dstct colourlaplacaege values of coloured graph wth ther multplctes are m 1,m,...m r s defed as colourlaplaca spectrum of a graph G s wrtte as LSpec c(g)={ μ 1, μ,. μ r m 1, m,. m r } color spectrum of G. III Results A. Lemma: If {λ 1, λ } s the colour spectrum of a k-regular graph G, the { k λ, k λ -1, k λ 1} s the colourlaplaca spectrum of G. B. Lemma:If the graph G s regular, the LE C (G) = E C (G). C. Lemma: For, the colour Laplaca Eergy of complete graph s (-1). D. Lemma:If {λ 1, λ,...,λ } s the colour spectrum of a k- regular graph G, the {k +λ 1, k +λ,..., k +λ } s the coloursgless Laplaca spectrum of G. AIJRSTEM ; 018, AIJRSTEM All Rghts Reserved Page 09

3 Bb et al., Amerca Iteratoal Joural of Research Scece, Techology, Egeerg & Mathematcs, 3(1), Jue-August, 018, pp IV Relato betwee colourlaplaca eergy ad coloursglesslaplaca eergy of a complete graph. Each vertex colour of a complete graph are dfferet because they are adjacet wth each other, so the chromatc umber of complete graph s. Cosder complete graph K 5 whch s gve below: The colourlaplaca matrx of k 5 s as follows: L C(k 5) = Fg() G=K5 The characterstc equato s μ(µ 5) 5 = 0. The ege values of k 5 are 0, 5, 5, 5, 5,5. The coloursglesslaplaca matrx for complete graph K 5 s L + C (k 5) = The Characterstc equato s (µ-3) 4 (µ-8)=0 The ege values are µ + = 3(4 tmes), µ + = 8 cosder the followg graph K 6 : Fg() G=k6 AIJRSTEM ; 018, AIJRSTEM All Rghts Reserved Page 10

4 Bb et al., Amerca Iteratoal Joural of Research Scece, Techology, Egeerg & Mathematcs, 3(1), Jue-August, 018, pp The colourlaplaca matrx of k 6 as follows : L C(k 6) = The characterstc equato s μ(μ 6) 5 =0 The ege values are 0, 6, 6, 6, 6, 6 The coloursglesslaplaca matrx of k 6 s as follows : L C+ (k 6) = The Characterstc equato s (µ - 4) 5 (µ -10) = 0 The ege values are µ = 4 (5 tmes), µ=10 Cotug the above process, the colourlaplaca matrx of complete graphk s L C(k ) = The characterstc equato s sμ(μ ) 1 = 0. The ege values are 0, (-1) tmes The sglesscolourlaplaca matrx of complete graph k s as follows : L + C (k 6) = The characterstc equato s (µ - (-1)) -1 (µ - (-1)) = 0 The ege values are [-1](-1) tmes, (-1) Hece the average degree of complete graph s -1. The Colour Laplaca Eergy of complete graph s L C(k ) = 0 ( 1) + ( 1) ( 1) = = - = ( - 1). L C+ (k ) = ( ) ( 1) ( 1) + ( 1) = AIJRSTEM ; 018, AIJRSTEM All Rghts Reserved Page 11

5 Bb et al., Amerca Iteratoal Joural of Research Scece, Techology, Egeerg & Mathematcs, 3(1), Jue-August, 018, pp = = ( 1). Thus, the colour Laplaca Eergy ad sgless color Laplaca Eergy of the complete graph k are the same ad t s equal to (-1). V. Bouds for colour eergy of graphs: I ths secto, we preseted some ew bouds for the colour eergy of a graph terms of Zagreb dex Z g(g), laplaca eergy LE(G) ad the sglesslaplaca eergy LE + (G). Bouds for colour eergy terms of zagreb dex: For a graph G of order ad sze m havg d as the degree of the th vertex We kow that =1 d = m (1) the Zagreb dex s Z g (G) = =1 d () Bouds for colour eergy terms of laplaca eergy ad sglesslaplaca eergy: I ths secto, we preseted some bouds for colour eergy terms of laplaca eergy ad the sglesslaplaca eergy. The followg results proved by Abreuetal.(see [1]). LE + (G) E(G) + Z g (G) 4m (3) LE + (G) LE(G) E(G) (4) 5.1 Theorem Let G be a graph wth order ad sze m. Let σ 1 σ σ be the absolute colour laplaca ege values of G. If σ 1 s repeated (-1) tmes the σ 1 1 ( ( 1 ) + (d 1 (G)) ( 1) =1 ( 1) = σ ) 1 Proof: Here we compare the absolute colourlaplacaege values of G wth the absolute ege values of the graph H=( 1 ). ( 1) (, 1 ) Select ad 1 such that =(-1)[+( 1)]. The umber of vertces of H s ad the umber of edges s (-1)()( 1 ). Its absolute values of ege values spectrum s By Cauchy s Schwarz equalty, σ 1 ( 1) + + σ 1 ( 1) But σ 1 = σ = = σ 1 0 ) ( 1) ( ( 1)) ( ( 1) + σ ( 1) + + σ ( 1) ( 1) m + =1 + σ 1 (0) +σ (0) (d (G)) )( 1)()( 1 σ 1 ( 1) ( 1) + ( 1) ( 1) = σ m + (d (G)) )( 1)()( 1 =1 ( 1) σ σ 1 ( 1) + = m + =1 (d (G)) )( 1) σ 1 1 m + (d 1 (G)) ( 1) =1 )( 1) = 1 σ ) ) AIJRSTEM ; 018, AIJRSTEM All Rghts Reserved Page 1

6 Bb et al., Amerca Iteratoal Joural of Research Scece, Techology, Egeerg & Mathematcs, 3(1), Jue-August, 018, pp Refereces [1] N. Abreu et al., Bouds for the sgless Laplaca eergy, Lear Algebra ad ts Applcatos (011). [] C. Adga et al., Color eergy of a graph, Proc. Jagjeo Math. Soc., 16 (013), [3] Balakrsha. R, The eergy of a graph, Lear Algebra ad ts applcatos(004) [4] Bapat R. B.,Pat S, Eergy of a graph s ever a odd teger. Bull.Kerala Math. Assoc. 1, (011). [5] Bo Zhou, Eergy of a graph, MATCH commu. Math. Chem. 51(004), [6] Bo Zhou ad Iva Gutma, O Laplaca eergy of a graph, Match,commu. Math. Comput. Chem. 57(007) [7] Bo Zhou, More o Eergy ad Laplaca Eergy Math. Commu. Math.,Comput. Chem.64(010) [8] Bo Zhou ad Iva Gutma, O Laplaca eergy of a graph, Lear algebra ad ts applcatos, 414(006) [9] G. Chartrad ad P. Zhag, Chromatc Graph Theory, CRC Press, New York (009). [10] D. M. Cvetkov c et al., Spectra of Graphs- Theory ad Applcato, Academc Press, New York (1980). [11] Cvetkov c.d, Doob M, Sachs H, Spectra of graphs - Theory ad applcatos, Academc Press, New York [1] FathTabar G.H, Ashraf A.R. I. Gutma, Note o Laplaca Eergy of graphs, class Bullet T(XXXVII de l AcademcsSerbe desscec,etdesarts(008). [13] Germa K.A, Shahul Hameed K, Thomas Zaslavsky, O products ad le graphs, ther egevalues ad eergy, Lear Algebra ad ts applcatos 435(011) [14] Gholam Hosse, Fath - Tabar ad Al Reza Asharf, Some remarks o Laplaca ege values ad Laplaca eergy of graphs, Math. [15] I. Gutma, The eergy of a graph, Ber. Math. Stat. Sekt. Forschugsz. Graz, 103, (1978), 1-. [16] G. Idulal et al., O dstace eergy of graphs, MATCH Commu. Math. Comuput.,Chem., 60 (008), [17] P. B. Josh ad M. Joseph ( press). Further results o color eergy of graphs. Act, UverstatsSapetae, Iformatca. [18] pradeepg.bhat ad Sabtha D Souza Colourlaplaca eergy of a graphs. Proceedgs of the jageo mathematcal socety, 18(015),No,3 pp [19] Srdhara G, M.R.Rajeshkama, bouds o eergy ad laplaca eergy of graphs, Math. [0] X. L et al., Graph Eergy, Sprger, New York (01). [1] D. B. West, Itroducto to Graph Theory, Pearso, New Jersey (001). AIJRSTEM ; 018, AIJRSTEM All Rghts Reserved Page 13