Double Dominating Energy of Some Graphs

Transcription

1 Iter. J. Fuzzy Mathematcal Archve Vol. 4, No., 04, -7 ISSN: (P), (ole) Publshed o 5 March 04 Iteratoal Joural of V.Kaladev ad G.Sharmla Dev P.G & Research Departmet of Mathematcs, Seethalakshm Ramaswam College Trchy-, Ida, Emal: kaladev956@gmal.com Departmet of Mathematcs, Kogu Arts ad Scece College,Erode,Ida. Research Scholar, Research ad Developmet Cetre, Bharathar Uversty, Combatore, Ida Emal: sharmlashamrtha@gmal.com Receved February 04; accepted March 04 Abstract. I ths paper the double domatg eergy of a graph s troduced. The double domatg eergy of a crow graph, cocktal party graph ad complete graph are computed. Keywords: Eergy, double domatg set, double domatg matrx, double domatg eergy. AMS Mathematcs Subect Classfcato (00): 05C0. Itroducto The cocept of eergy of a graph was troduced by I. Gutma [], the year 978. Let G be a graph wth vertces ad m edges ad let A = (a ) be the adacecy matrx of the graph. The ege values λ, λ, λ 3,..., λ of A, assumed o-decreasg order λ λ λ 3... λ are the ege values of the graph G. Sce A(G) s real ad symmetrc, ts ege values are real umbers. The eergy E(G) of G s defed to be the sum of the absolute values of ts ege values of G. That s E(G) = λ. I HMO Theory, the total eergy of the π electros s equal to the sum of the eerges of allπ -electros the cosdered molecule. It ca be calculated from the ege values of the uderlyg molecular graph [, 8]. Smlar to eerges lke Laplaca eergy, dstace eergy, mmum coverg eergy, cdece eergy [3,4, 5, 6, 7], the double domatg eergy s defed ths paper ad same s foud out for some graphs.. Double domatg eergy Let G be a smple graph of order wth vertex set V = {v, v, v 3,...,v } ad edge set E. A subset D V s a double domatg set f D s a domatg set ad every vertex of V D s adacet to atleast two vertces D. The Double Domato umber γ x (G) s the mmum cardalty take over all the mmal double domatg sets of G.

2 V.Kaladev ad G.Sharmla Dev Let D be the mmum double domatg set of a graph G. The mmum double domatg matrx of G s the matrx defed by A D (G) = ( a ) where f vv E a = f = ad v D 0 otherwse The characterstc polyomal of A D (G) s deoted by f(g, λ) = det(λi A D (G). The mmum double domatg ege values of the graph G are the ege values of A D (G). Sce A D (G) s real ad symmetrc, ts ege values are real umbers ad are labelled o-creasg order λ λ λ 3... λ. The mmum double domatg eergy of G s defed as E D (G) = λ. Example.. Let G be a cycle C 4 o 4 vertces u, u, u 3, u 4 wth mmum double domatg set D = {u, u 3 }. The A D (C 4 ) = The characterstc polyomal of A D (C 4 ) s λ 4 λ 3 3λ + 4λ, the mmum double domatg ege values are 0,, + 7 7,, ad the mmum double domatg eergy s E D (C 4 ) = Propertes of double domatg eergy Theorem 3.. Let G be a graph wth vertces ad m edges. If λ, λ, λ 3,..., λ are the ege values of A D (G), the λ = E + D. Proof : The sum of square of the ege values of A D (G) s the trace of A D (G). λ = aa = = ( a ) + ( a ) < = E + D = m + D. Theorem 3.. Let G be a smple graph wth vertces, m edges ad let D be a double domatg set of G ad F = deta D (G) the / m + D + (-)f E (G) (m + D ) Proof : Let λ λ λ 3... D λ be the ege values of A D (G). By Cauchy-Schwarz equalty,

3 ( ) ( ab ) ( a ) ( b ) leta, b = λ, E D (G) = ( λ ) ( λ ) = E D (G) (m + D [E D (G)] = ( λ ) = (m + D + λ λ λ From the equalty betwee the arthmetc ad geometrc mea, we obta ( λ λ ) λ λ λ λ ( )F λ λ ( ) ( ) λ ( ) ( ) ( ) λ ( ) λ ( ) λ ( ) deta (G) [E D (G)] / λ + ( - )F [E D (G)] D (m + D + ( - )F (m + D + ( - )F Bapat ad Pat showed that f the graph eergy s a ratoal umber, the t s a eve teger [9]. The aalogous result for mmum double domatg eergy s gve the followg theorem. Theorem 3.3. Let G be a graph wth a mmum double domatg set D. If the mmum double domatg eergy E D (G) s a ratoal umber, thee D (G) = D (mod). Proof: Let λ, λ, λ 3,...,λ be the mmum double domatg ege values of a graph G of whch λ, λ,...,λ r are postve ad the remag are o-postve, the / / 3

4 V.Kaladev ad G.Sharmla Dev E D (G) = (λ + λ λ r ) (λ r λ ) = (λ + λ λ r ) (λ + λ λ ) = (λ + λ λ r ) - λ = (λ + λ λ r ) - D E D (G) = D (mod) 4. Double domatg eerges of some famles of graphs Defto 4.. The crow graph S for a teger s the graph wth vertex set{u, 0 u,..... u, v, v,..... v } ad the edge set {u v :,, }. Theorem 4.. For 4, the double domatg eergy of the crow graph S o s equal to + ( 3) Proof: 0 The crow graph S wth vertex set v = {u, u,..... u, v, v,..... v }, the mmum double domatg set D = {u, u, v, v,} The AD ( S) = characterstc polyomal s λ λ λ λ 0 0 λ λ λ λ characterstc equato s 4

5 3 3 λ(λ )(λ ) + (λ ) (λ + ( 3))λ ( 4))(λ ( )λ ) = 0 Mmum double domatg ege values are 0,, -, ( 3 tmes), ( 3 tmes) ( ) ± + 9 (3 ) ± + 7 (oe tme each) (oe tme each) Mmum double domatg eergy 0 E (S ) = + ( - 3) D Theorem 4.3. The double domatg eergy of the complete graph K s equal to ( - 3) Proof: The complete graph K wth vertex set v = {v, v,.....v }, the mmum double domatg set D = {v, v }. The 0 AD ( K) = characterstc polyomal s λ λ λ λ λ characterstc equato s 3 λ(λ + ) (λ ( )λ ) = 0 Mmum double domatg ege values are 0, - ( 3 tmes) ( ) ± + 9 (oe tme each) Mmum double domatg eergy E (K ) = ( - 3) D 5

6 V.Kaladev ad G.Sharmla Dev Defto 4.4. The cocktal party graph s deoted by K, s a graph havg the vertex set V = { u, v } ad the edge set E = { uu, vv : } { uv, vu : < } Theorem 4.5. The mmum double domatg eergy of cocktal party graph K x s ( - 3) Proof: Let K x be the cocktal party graph wth vertex set V = { u, v }. The mmum double domatg set s D = {u, v }. The AD ( K ) = characterstc polyomal s λ 0 0 λ λ 0 0 λ λ 0 0 λ λ 0 0 λ characterstc equato s λ (λ )(λ + ) (λ ( 3)λ ) = 0 Mmum double domatg ege values are 0( tmes),, -(- tmes) 6

7 ( 3) ± Mmum double domatg eergy (oe tme each) E D (K ) = + ( ) = ( - 3) Cocluso Thus ths paper, the ew eergy amely the double domatg eergy s defed ad has bee foud for some graphs. REFERENCES. I. Gutma, The eergy of a graph, Ber. Math-Statst.Sekt. for Schugsz. Graz, 03 (978) -.. A. Graovac, I. Gutma ad N. Trastc, Topologcal approach to the chemstry of cougated molecules, Sprger-Verlag, Berl, I. Gutma, B. Zhou, Laplacea eergy of a graph, L. Algebra Appl., 44 (006) G. Idulal, I. Gutma ad A. Vayakumar, O dstace eergy of graph, MATCH Commu. Math. Comput. Chem., 60 (008) C. Adga, A. Bayad, J.Gutma, S.A. Srvas, The mmum coverg eergy of a graph, Kraguevac J. Sc., 34 (0) M.R. Joo, D. Yadeh ad M. Ka, Mrzakhah, Icdece eergy of a graph, MATCH commu. Math. Comput. Chem., 6 (009) J. Lu, B. Lu, A Laplacea eergy lke varat of a graph, MATCH Commu. Math. Comput. Chem., 59 (008) I. Gutma, Topologcal studes o hetero cougated molecules. alterat systems wth oe heteroatom, Theor. Chm. Acta, 50 (979) R.B.Bapat ad S.Pat, Eergy of a graph s evera odd teger, Bull. Kerala Math. Asscc., (0)