Minimum Equitable Dominating Randić Energy of a Graph

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1 Iteratioal JMath Combi Vol28), 97-8 Miimum Equitable Domiatig Raić Eergy of a Graph Rajera P Bharathi College, PG a Research Cetre, Bharathiagara, , Iia) RRagaraja DOS i Mathematics, Uiversity of Mysore, Mysuru, 57 6, Iia) prajumaths@gmailcom, rajra6@gmailcom Abstract: Let G be a graph with vertex set V G), ege set EG) a i is the egree of its i-th vertex v i, the the Raić matrix RG) of G is the square matrix of orer, whose i, j)-etry is equal to if the i-th vertex v i a j-th vertex v j of G are ajacet, i j a zero otherwise The Raic eergy [] REG) of the graph G is efie as the sum of the absolute values of the eigevalues of the Raić matrix RG) A subset ED of V G) is calle a equitable omiatig set [], if for every v i V G) ED there exists a vertex v j ED such that v iv j EG) a iv i) jv j) I the cotrast, such a omiatig set ED is Smaraachely if iv i) jv j) Recetly, Aiga, etal itrouce, the miimum coverig eergy EcG) of a graph [] a S Burcu Bozkurt, etal itrouce, Raić Matrix a Raić Eergy of a graph [] Motivate by these papers, Miimum equitable omiatig Raić eergy of a graph RE EDG) of some graphs are worke out a bous o RE EDG) are obtaie Key Wors: Raić matrix a its eergy, miimum equitable omiatig set a miimum equitable omiatig Raić eergy of a graph AMS2): 5C5 Itrouctio Let G be a graph with vertex set V G) a ege set EG) The ajacecy matrix AG) of the graph G is a square matrix, whose i, j)-etry is equal to if the vertices v i a v j are ajacet, otherwise zero [2] Sice AG) is symmetric, its eigevalues are all real Deote them by λ, λ 2,,λ, a as a whole, they are calle the spectrum of G a eote by SpecG) The eergy of graph [2] G is εg) = i= λ i The literature o eergy of a graph a its bous ca refer [4,8,9,,2] The Raić matrix RG) = r ij ) of G is the square matrix of orer, where Receive July 28, 27, Accepte March, 28

2 98 Rajera P a RRagaraja r ij ) =, if v i a v j are ajacet vertices i G; i j, otherwise The Raic eergy [] REG) of the graph G is efie as the sum of the absolute values of the eigevalues of the Raić matrix RG) Let ρ, ρ 2,, ρ be the eigevalues of the Raić matrix RG) Sice RG) is symmetric matrix, these eigevalues are real umbers a their sum is zero Raić eergy [] ca be efie as REG) = i= ρ i For etails of Raić eergy a its bous, ca refer [2,, 5, 6, 7] Let G be a graph with vertex set V G) = {v, v 2,, v } a ED is miimum equitable omiatig set of G Miimum equitable omiatig Raić matrix of G is matrix R ED G) = r ij ), where, if v i a v j are ajacet vertices i G; i j r ij ) =, if i = j a v i ED;, otherwise The characteristic polyomial of R ED G) is eote by etρi R ED G)) = ρi R ED G) Sice R ED G) is symmetric, its eigevalues are real umbers If the istict eigevalues of R ED G) are ρ > ρ 2 > > ρ r with their multiplicities are m, m 2,, m r the spectrum of R ED G) is eote by SpecR ED G) = ρ ρ 2 ρ r m m 2 m r The miimum equitable omiatig Raić eergy of G is efie as RE ED G) = ρ i Example Let W 5 be a wheel graph, with vertex set V G) = {v, v 2, v, v 4, v 5 }, a let its miimum equitable omiatig set be ED = {v } The miimum equitable omiatig Raić matrix R ED W 5 ) is R ED W 5 ) = i= SpecR ED W 5 ) = 2

3 Miimum Equitable Domiatig Raić Eergy of a Graph 99 The miimum equitable omiatig Raić eergy of W 5 is RE ED W 5 ) = 22 2 Bous for the Miimum Equitable Domiatig Raić Eergy of a Graph Lemma 2 If ρ, ρ 2,, ρ are the eigevalues of R ED G) The a ρ i = ED i= i= ρ 2 i = ED + 2 where ED is miimum equitable omiatig set Proof i) The sum of eigevalues of R ED G) is ρ i = i=, r ii = ED i= ii) Cosier, the sum of squares of ρ, ρ 2,, ρ is, ρ 2 i = i= = r ij r ji = ii ) i= j= i=r 2 + r ij r ji i =j ii ) i=r r ij ) 2 i= ρ 2 i = ED + 2 ¾ Upper a lower bous for RE ED G) is similar proof to McClella s iequalities [], are give below Theorem 22Upper Bou) Let G be a graph with ED is miimum equitable omiatig set The RE ED G) ED + 2 i j Proof Let ρ, ρ 2,, ρ be the eigevalues of R ED G) By Cauchy-Schwartz iequality, we have ) 2 ) ) a i b i, ) where a a b are ay real umbers a 2 i b 2 i i= i= i=

4 Rajera P a RRagaraja If a i =, b i = ρ i i ), we get 2 ) ) ρ i ) 2 ρ i 2, [RE ED G)] 2 ED + 2 i= i= i= by Lemma 2, RE ED G) ED + 2 Theorem 2Lower Bou) Let G be a graph with ED is miimum equitable omiatig set a i is egree of v i The where D = i= ρ i RE ED G) ED )D 2, Proof Cosier [ 2 [RE ED G)] 2 = ρ i ] = By usig arithmetic a geometric mea iequality, we have ) ρ i ρ j i =j i =j i= ρ i 2 + ρ i ρ j 2) i =j i= ρ i ρ j i =j i= ) ) ) ρ i ρ j ) ρ i 2) 2/ ρ i ρ j ) ρ i ) ) i =j Now usig ) i 2), we get [RE ED G)] 2 i= 2/ ρ i 2 + ) ρ i ), i= [RE ED G)] 2 ED )D 2, where D = ρ i, i j i= RE ED G) ED )D 2 i j i=

5 Miimum Equitable Domiatig Raić Eergy of a Graph Bous for Largest Eigevalue of R ED G) a its Eergy Propositio Let G be a graph a ρ G) = max i { ρ i } be the largest eigevalue of R ED G) The ED + 2 ρ G) ED + 2 i j Proof Cosier, ρ 2 G) = max i { ρ i 2 } ρ G) ED + 2 i j ρ i 2 = ED + 2 i= Next, ρ 2 G) ρ 2 G) ρ G) ρ 2 i = ED + 2 i= i j ED + 2 i j ED + 2 i j Therefore, ED + 2 ρ G) ED + 2 Propositio 2 If G is a graph a ED + 2 RE EDG) ED ) ED + 2 i j, the i j ED + 2 ) 2 Proof We kow that, i= ρ 2 i = ED + 2, i=2 ρ 2 i = ED + 2 ρ 2 4)

6 2 Rajera P a RRagaraja By Cauchy-Schwarz iequality, we have ) 2 ) ) a i b i a 2 i b 2 i i=2 i=2 i=2 If a i = a b i = ρ i, we have 2 ρ i ) ) ρ i 2 i=2 i=2 [RE ED G) ρ ] 2 ) ED + 2 ρ 2 i j RE ED G) ρ + ) ED + 2 ρ 2 5) i j Cosier the fuctio, Fx) = x + ) ED + 2 x i 2 j The, F x) = x ) ED + 2 i j x 2 Here Fx) is ecreasig i ED + 2, ED + 2 We kow that F x), x ) ED + 2 i j x 2 We have Sice, x ED + 2 i j ED + 2 a ED + 2 i j i j ρ,

7 Miimum Equitable Domiatig Raić Eergy of a Graph we have ED + 2 i j ED + 2 i j ρ ED + 2 The, equatio 5) become RE ED G) ED + 2 i j + ) ED + 2 ED + 2 i j ) 2 ¾ 4 Miimum Equitable Domiatig Raić Eergy of Some Graphs Theorem 4 If K is complete graph with vertices, the miimum equitable omiatig Raić eergy of K is RE ED K ) = 5 Proof Let K be the complete graph with vertex set V G) = {v, v 2,,v } a miimum equitable omiatig set is ED = {v }, we have characteristic polyomial of R ED K ) is ρ ρi R ED K ) = ρ ρ ρ ρ R k = R k R 2, k =, 4,,, The, we get a we have ρi R ED K ) = ρ + ) 2 ρ ) ρ + commo from R to R ρ ρ

8 4 Rajera P a RRagaraja C 2 = C 2 + C + + C, we get the characteristic polyomial ρi R ED K ) = SpecR ED K ) = ρ + ) 2 [ ρ 2 2 ) 4 2) ) 2 ρ + 2 )+ 4 2) 2 The miimum equitable omiatig Raić eergy of K is )], RE ED K ) = 5 ¾ Theorem 42 If S, 4) is star graph with vertices, the miimum equitable omiatig Raić eergy of S is RE ED S ) = Proof Let S, 4) be the star graph with vertex set V G) = {v, v 2,, v } a miimum equitable omiatig set is ED = {v, v 2,, v }, we have characteristic polyomial of R ED S ) is ρ ρ ρ ρi R ED S ) = ρ ρ ρ R k = R k R 2, k =, 4,, The takig ρ ) commo from R to R, we get ρ ρ ρi R ED S ) = ρ ) 2 C 2 = C 2 + C + + C The, the characteristic polyomial ρi R ED S ) = ρρ ) 2 ρ 2),

9 Miimum Equitable Domiatig Raić Eergy of a Graph 5 2 SpecR ED S ) = 2 The miimum equitable omiatig Raić eergy of S is RE ED S ) = ¾ Theorem 4 If K m,, where m < a m 2 is complete bipartite graph with m + vertices, the miimum equitable omiatig Raić eergy of K m, is RE ED K m, ) = m+ Proof Let K m,, where m < a m 2 be the complete bipartite graph with vertex set V G) = {v, v 2,, v m, u, u 2,, u } a miimum equitable omiatig set is ED = V G), we have characteristic polyomial of R ED K m, ) is ρ ρ ρ ρ ρi R ED K m, ) = ρ ρ ρ ρ R k = R k R m, k =, 2,,, m a R = R R m+, = m + 2, m +,, m + The, takig ρ ) commo from R to R m a R m+2 to R m+, we get ρ ) m+ 2 ρ m m m m m m m m ρ

10 6 Rajera P a RRagaraja C m+ = C m+ + C m C m+, ρ ) m+ 2 ρ m m m m m m m m ρ The characteristic polyomial ρi R ED K m, ) = ρ ρ ) m+ 2 ρ 2), SpecR ED K m, ) = 2 m + 2 The miimum equitable omiatig Raić eergy of K m, is RE ED K m, ) = m + ¾ Theorem 44 If K 2, ) is cocktail party graph with 2 vertices, the miimum equitable omiatig Raić eergy of K 2 is 4 6 Proof Let K 2, ) be the cocktail party graph with vertex set V G) = {v, v 2,, v, u, u 2,, u } a miimum equitable omiatig set is ED = {v, u }, we have characteristic polyomial of R ED K 2 ) is λ λ ρi R EDK 2) = λ λ λ λ λ λ

11 Miimum Equitable Domiatig Raić Eergy of a Graph 7 R k = R k R, k = 4, 5,, a R +k = R +k R k+, k = 2,,,, we get λ λ λ λ λ + ρi R EDK 2) = λ λ + λ λ λ λ λ C = C + C C + C C 2 a C k = C k + C +k), k = 4, 5,, + We get λ λ λ 2 4) ) λ + ρi R EDK 2) = λ + λ λ λ The characteristic polyomial ρi R ED K 2 ) = ρ ρ ) SpecR ED K 2 ) = ρ + ) 2 [ ρ 2 2 ) 4 2) ) 2 ρ + 2 The miimum equitable omiatig Raić eergy of K 2 is 2 )+ 4 2) )], where RE ED K 2 ) = 4 6, ¾

12 8 Rajera P a RRagaraja Refereces [] C Aiga, A Baya, I Gutma a Shrikath A S, The miimum coverig eergy of a graph, Krag J Sci, 4 22), 9-56 [2] Ş Burcu Bozkurt, A Dilek Gügör a Iva Gutma, Raić, Spectral raius a Raić eergy, MATCH Commu Math Comput Chem, 64 2), 2-4 [] Ş Burcu Bozkurt, A Dilek Gügör, Iva Gutma a A Sia Çevik, Raić matrix a Raić eergy, MATCH Commu Math Comput Chem, 64 2), [4] D M Cvetkovic, M Doob a H Sachs, Spectra of Graphs, Theory a Applicatio, Acaemic Press, New York, USA, 98 [5] K C Das a S Sorgu: O Raić eergy of graphs, MATCH Commu Math Comput Chem, 72 24), [6] A Dilek Mae, New bous o the iciece Eeergy, Raić eergy a Raić Estraa iex, MATCH Commu Math Comput Chem, 74, 25), [7] B Furtula a I Gutma, Comparig eergy a Raić eergy, Maceoia Joural of Chemistry a Chemical Egieerig, 2), 2), 7-2 [8] JH Koole a V Moulto, Maximal eergy graphs Avaces i Applie Mathematics, 26, 2), [9] I Gutma, The eergy of a graph, Ber Math-Statist Sekt Forschugsz Graz, 978), -22 [] B J McClella, Properties of the latet roots of a matrix: The estimatio of π-electro eergies, J Chem Phys, 54 97), [] V Swamiatha a K M Dharmaligam, Degree equitable omiatio o graphs, Krag J of Math, 5 ) 2), 9-97 [2] X Li, Y Shi a I Gutma, Graph Eergy, Spriger New York, 22