Laplacian Minimum covering Randić Energy of a Graph

Transcription

1 Commuicatios i Mathematics ad Applicatios Vol 9 No pp ISSN (olie); (prit) Published by RGN Publicatios Laplacia Miimum coverig Radić Eergy of a Graph MR Rajesh Kaa * ad R Jagadeesh Research Article Sri D Devaraj Urs Govermet First Grade College Husur 57 5 Mysore District Karataka Idia Research ad Developmet Ceter Bharathiar Uiversity Coimbatore Idia Govermet Sciece College (Autoomous) Nrupathuga Road Bagalore 56 Idia *Correspodig author: mrrajeshkaa@gmailcom Abstract Radić eergy was first defied i the article [6] Usig miimum coverig set we have itroduced i this article Laplacia miimum coverig Radić eergy LRE C (G) of a graph G This article cotais computatio of Laplacia miimum coverig Radić eergies for some stadard graphs like star graph complete graph crow graph complete bipartite graph ad cocktail graph At the ed of this article upper ad lower bouds for Laplacia miimum coverig Radić eergy are also preseted Keywords Miimum coverig set; Miimum coverig Radić matrix; Laplacia miimum coverig Radić matrix; Laplacia miimum coverig Radić eigevalues; Laplacia miimum coverig Radić eergy MSC Primary 5C5 5C69 Received: December 6 Accepted: November 7 Copyright 8 MR Rajesh Kaa ad R Jagadeesh This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited Itroductio Study o eergy of graphs goes back to the year 978 whe Gutma [6] defied this while workig with eergies of cojugated hydrocarbo cotaiig carbo atoms All graphs cosidered i this article are assumed to be simple without loops ad multiple edges Let A = (a i j ) be the adjacecy matrix of the graph G with its eigevalues ρ ρ ρ ρ assumed i decreasig order Sice A is real symmetric the eigevalues of G are real umbers whose sum equal

2 68 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh to zero The sum of the absolute eigevalues values of G is called the eergy E(G) of G ie E(G) = ρ i i= Theories o the mathematical cocepts of graph eergy ca be see i the reviews [] articles [9 7] ad the refereces cited there i For various upper ad lower bouds for eergy of a graph ca be foud i articles[ ] ad it was observed that graph eergy has chemical applicatios i the molecular orbital theory of cojugated molecules [5 9] Gutma ad Zhou [] defied the Laplacia eergy of a graph G i the year 6 Let G be a graph with vertices ad m edges The Laplacia matrix of the graph G deoted by L = (L i j ) is a square matrix of order whose elemets are defied as if v i ad v j are adjacet L i j = if v i ad v j are ot adjacet d i if i = j where d i is the degree of the vertex v i Let µ µ µ be the Laplacia eigevalues of G Laplacia eergy LE(G) of G is defied as LE(G) = µi m The basic properties icludig various upper ad lower bouds for Laplacia eergy have bee established i [ ] ad it has foud remarkable chemical applicatios the molecular orbital theory of cojugated molecules [8] Radić Eergy It was i the year 975 Mila Radić iveted a molecular structure descriptor called Radić idex which is defied as [9]: R(G) = di d j v i v j E(G) Motivated by this Bozkurt et al [6] defied Radić matrix ad Radić eergy as follows Let G be graph of order with vertex set V = {v v v } ad edge set Radić matrix of G is a symmetric matrix defied by R(G) := (r i j ) where if v i v j E(G) r i j = di d j () otherwise The characteristic equatio of R(G) is defied by f (Gρ) = det(ρi R(G)) = The roots of this equatio is called Radić eigevalues of G Sice R(G) is real ad symmetric its eigevalues are real umbers ad we label them i decreasig order ρ ρ ρ Radić eergy of G is defied as RE(G) := ρ i i= Further studies o Radić eergy ca be see i the articles [7 5] ad the refereces cited therei Miimum Coverig Eergy I the year Adiga et al [] itroduced miimum coverig eergy of a graph which depeds o its particular miimum cover A subset C of vertex set V is called a coverig set of G if every edge of G is icidet to at least oe vertex of C Ay coverig set with miimum cardiality is i= Commuicatios i Mathematics ad Applicatios Vol 9 No pp

3 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 69 called a miimum coverig set If C is a miimum coverig set of a graph G the the miimum coverig matrix of G is the matrix defied by A C (G) := (a i j ) where if v i v j E(G) a i j = if i = j ad v i C () otherwise The miimum coverig eigevalues of the graph G are roots of the characteristic equatio f (Gρ) = = det(ρi A C (G)) = obtaied from the matrix A C (G) Sice A C (G) is real ad symmetric its eigevalues are real umbers ad we label them i the order ρ ρ ρ The miimum coverig eergy of G is defied as E C (G) := ρ i Miimum coverig Radić Eergy Results o Radić eergy ad miimum coverig eergy of graph G motivates us to defie miimum coverig Radić eergy Cosider a graph G with vertex set V = {v v v } ad edge set E If C is a miimum coverig set of a graph G the the miimum coverig Radić matrix of G is the matrix defied by R C (G) := (r i j ) where if v i v j E(G) di d j r i j = if i = j ad v i C otherwise The characteristic polyomial of R C (G) is defied by f (Gρ) = det(ρi R C (G)) The miimum coverig Radić eigevalues of the graph G are the eigevalues of R C (G) Sice R C (G) is real ad symmetric matrix so its eigevalues are real umbers We label the eigevalues i order ρ ρ ρ The miimum coverig Radić eergy of G is defied as RE C (G) := ρ i I the year Adiga et al [] itroduced miimum coverig eergy of a graph which depeds o its particular miimum cover Motivated by this article recetly Rajesh Kaa ad Jagadeesh i the article [8] itroduced the cocept of miimum coverig Radić eergy 4 Laplacia Eergy Gutma ad Zhou i article [] itroduced the Laplacia eergy of a graph G i the year 6 Defiitio Let G be a graph with vertices ad m edges The Laplacia matrix of the graph G deoted by L = (L i j ) is a square matrix of order whose elemets are defied as if v i ad v j are adjacet L i j = if v i ad v j are ot adjacet if i = j d i where d i is the degree of the vertex v i Defiitio Let µ µ µ be the Laplacia eigevalues of G Laplacia eergy LE(G) of G is defied as LE(G) = µi m i= i= i= Commuicatios i Mathematics ad Applicatios Vol 9 No pp

4 7 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 5 Laplacia Miimum coverig Eergy of A Graph Let D(G) be the diagoal matrix of vertex degrees of the graph G The L C (G) = D(G) A C (G) is called the Laplacia coverig matrix of G Let µ µ µ be the eigevalues of L C (G) arraged i o-icreasig order These eigevalues are called Laplacia miimum coverig eigevalues of G The Laplacia miimum coverig eergy of the graph G is defied as LE C (G) := µ i m i= where m is the umber of edges of G ad m is the average degree of G 6 Laplacia Miimum Coverig Radić Eergy of A Graph Let D(G) be the diagoal matrix of vertex degrees of the graph G The LR C (G) = D(G) R C (G) is called the Laplacia coverig matrix of G Let ρ ρ ρ ρ be the eigevalues of LR C (G) arraged i o-icreasig order These eigevalues are called Laplacia miimum coverig Radić eigevalues of G The Laplacia miimum coverig Radić eergy of the graph G is defied as LRE C (G) := ρ i m i= where m is the umber of edges of G ad m is the average degree of G I this article we are iterested i studyig mathematical aspects of the Laplacia miimum coverig eergy of a graph The applicatio of Laplacia miimum coverig eergy i other braches of sciece have to be ivestigated Example (i) C = {v v 4 v 6 } (ii) C = {v v 4 v 5 } are the possible miimum coverig sets for the Figure as show below Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 7 v v v v 4 v 5 v 6 Figure (i) R C (G) = D(G) = 4 Commuicatios i Mathematics ad Applicatios Vol 9 No pp Figure

5 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 7 LR C (G) = D(G) R C (G) = Laplacia miimum coverig Radić eigevalues are ρ 7 ρ 864 ρ ρ ρ ρ Number of vertices = 6 umber of edges = 7 Therefore average degree = m = 7 6 = 7 Laplacia miimum coverig Radić eergy LRE C (G) (ii) R C (G) = LR C (G) = D(G) R C (G) = 6 D(G) = Laplacia miimum coverig Radić eigevalues are ρ 595 ρ ρ 8894 ρ ρ ρ Number of vertices = 6 umber of edges = 7 Therefore average degree = m = 7 6 = 7 Laplacia miimum coverig Radić eergy LRE C (G) Therefore Laplacia miimum coverig Radić eergy depeds o the coverig set Mai Results ad Discussio aplacia Miimum Coverig Radić Eergy of Some Stadard Graphs Theorem For Laplacia miimum coverig Radić eergy of complete graph K is ( ) Commuicatios i Mathematics ad Applicatios Vol 9 No pp

6 7 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh Proof Let K be a complete graph with vertex set V = {v v v v } The Laplacia miimum coverig set for K is C = {v v v v } The R C (K ) = D(K ) = LR C (K ) = D(K ) R C (K ) = Characteristic polyomial is ( ) [( )ρ ( + )] [( )ρ ( 6 + 5)ρ + ( )] ( ) Characteristic equatio is ( ) [( )ρ ( + )] [( )ρ ( 6 + 5)ρ + ( )] ( ) = Laplacia miimum coverig Radić spec LR c (K ) = + ( 6+5) ( ) ( 6+5) ( ) Number of vertices = umber of edges = C = ( ) Therefore average degree = m = ( ) = Laplacia miimum coverig Radić eergy LRE C (K ) = + ( ) ( ) + ( 6 + 5) ( ) ( ) () Commuicatios i Mathematics ad Applicatios Vol 9 No pp

7 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 7 + ( 6 + 5) ( ) () ( ) = + ( ) ( ) + ( ) = ( ) + = ( ) () + ( ) ( ) Defiitio Cocktail party graph is deoted by K is a graph havig the vertex set V = {u i v i } ad the edge set E = {u i u j v i v j : i j} {u i v j v i u j : i < j } i= Theorem Laplacia miimum coverig Radić eergy of cocktail party graph K is ( ) Proof Cosider cocktail party graph K with vertex set V = miimum coverig set of cocktail party graph K is C = {u i v i } The R C (K ) = D(K ) = u u u u v v v v u u u u v v v v i= i= () {u i v i } The Laplacia u u u u v v v v u u u u v v v v Commuicatios i Mathematics ad Applicatios Vol 9 No pp

8 74 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh LR C (K ) = D(K ) R C (K ) u u u u v v v v u u u = u v v v v Characteristic polyomial is [ρ ( )][ρ ( )] [( )ρ (4 + 7)ρ + (4 6 + )] ( ) Characteristic equatio is [ρ ( )][ρ ( )] [( )ρ (4 + 7)ρ + (4 6 + )] ( ) = Laplacia miimum coverig Radić spec(k ) = ( (4 +7) ( ) (4 +7) ( ) Number of vertices = umber of edges = ( ) Therefore average degree = ()( ) = ( ) Laplacia miimum coverig Radić eergy LRE C (K ) = ( ) ( )() + ( ) ( ) ( ) + (4 + 7) ( ) + (4 + 7) ( ) = + ( ) =( ) + = ( ) ( ) () ( ) () ) () Theorem Laplacia miimum coverig Radić eergy of star graph is { 5 if = LR C (K ) = ( ) if > () Commuicatios i Mathematics ad Applicatios Vol 9 No pp

9 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 75 Proof Let K be a star graph with vertex set V = {v v v v } The its miimum coverig set is C = {v } Case (i): If = the characteristic equatio is ρ ρ = Laplacia miimum coverig Radić spec LR C (K ) = ( Number of vertices = umber of edges = Therefore average degree = () = Laplacia miimum coverig Radić eergy ) LRE C (K ) = + 5 () () = + = 5 Case (ii): If > R C (K ) = D(K ) = LR C (K ) = D(K ) R C (K ) = Characteristic equatio is ( ) [ρ ] [ρ ( )ρ ] = Laplacia miimum coverig Radić ( ( )+ 6+ spec LR C (K ) = ( ) 6+ ) Commuicatios i Mathematics ad Applicatios Vol 9 No pp

10 76 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh Laplacia miimum coverig Radić eergy ( ) LRE C (K ) = ( ) + ( ) ( ) 6 + = + ( ) () ( ) + ( 5 + 4) () + ( 5 + 4) 6 + () ( ) = ( ) () Defiitio Crow graph S for a iteger is the graph with vertex set {u u u v v v } ad edge set {u i v j : i j i j} Theorem 4 For Laplacia miimum coverig Radić eergy of the crow graph S is equal to Proof For the crow graph S with vertex set V = {u u u v v v } miimum coverig set of crow graph S is C = {u u u } The u u u u v v v v u u u R C (S ) = u v v v v u u u u v v v v u u u D(S ) = u v v v v Commuicatios i Mathematics ad Applicatios Vol 9 No pp

11 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 77 LR C (S ) = D(S ) R C(S ) u u u u v v v v u u u = u v v v v Characteristic polyomial is [ρ ( )ρ + ( + )][( ) ρ ( )( ) ρ + ( )] ( ) Characteristic equatio is [ρ ( )ρ + ( + )][( ) ρ ( )( ) ρ + ( )] ( ) = Laplacia miimum coverig Radić 5 ( ) 5 ( 5+)+ +5 ( ) ( 5+) +5 ( ) ( ( )+ spec S = Number of vertices = umber of edges = ( ) Therefore average degree = ( ) = Laplacia Miimum coverig Radić eergy LRE C (S ) = ( ) + 5 ( )() + ( ) 5 + ( 5 + ) ( ) ( )() ( )( ) + ( 5 + ) + 5 ( )( ) ( ) = () + () + ( + ) ( ) ( ) + ( + ) + 5 ( ) ( ) = Theorem 5 The miimum coverig Radić eergy R C (G) of the complete bipartite graph RE C (K m ) is equal to m m m(m ) (m ) + m + m + ( ) + (m + ) + m + m + 5 ) Commuicatios i Mathematics ad Applicatios Vol 9 No pp

12 78 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh Proof For the complete bipartite graph K m (m ) with vertex set V = {u u u m v v v } miimum coverig set is C = {u u u m } The u u u u m v v v v u m m m m u m m m m u m m m m R C (K m ) = u m m m m m v m m m m v m m m m v m m m m v m m m m u u u u m v v v v u u u D(K m ) = u m v m v m v m v m LR C (K m ) = D(K m ) R C (K m ) u u u u m v v v v u m m m u m m m u m m m = u m m m m v m m m m m v m m m m m v m m m m m m m m v Characteristic equatio is m m ( ) m+ [ρ ( ] m [ρ m] [ρ ( + m )ρ + (m (m + ))] = m m m m Commuicatios i Mathematics ad Applicatios Vol 9 No pp

13 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 79 Laplacia miimum coverig Radić ( spec+(k m ) = m (m+ )+ (m+)+m +m+5 m Number of vertices = m + umber of edges = m Therefore average degree = m m+ Laplacia miimum coverig Radić eergy (m+ ) (m+)+m +m+5 ) LRE C (K m ) = ( ) m m (m ) + m ( ) m + m + + (m + ) + (m + ) + m + m (m + ) (m + ) + m + m + 5 = m m m(m ) (m ) + ( ) m + m + m m + () m m + () + (m ) (m + ) + (m + ) (m + ) + m + m + 5 () (m + ) + (m ) (m + ) (m + ) (m + ) + m + m + 5 () (m + ) = m m m(m ) (m ) + ( ) m + m + + (m + ) + m + m + 5 Properties of Laplacia Miimum Coverig Radić Eigevalues Theorem 6 Let G be a graph with vertex set V = {v v v } edge set E ad C = {u u u k } be a miimum coverig set If ρ ρ ρ are the eigevalues of Laplacia miimum coverig Radić matrix LR C (G) the (i) ρ i = E C i= (ii) ρ i = M where M = i= (d i c i ) + i= i< j d i d j ad c i = { if vi C if v i C Proof (i) We kow that the sum of the eigevalues of LR C (G) is the trace of LR C (G) Therefore ρ i = r ii = d i C = E C i= i= i= (ii) Similarly the sum of squares of the eigevalues of LR C (G) is trace of [LR C (G)] Therefore ρ i = r i j r ji i= i= j= = ii ) i=(r + r i j r ji i j Commuicatios i Mathematics ad Applicatios Vol 9 No pp

14 8 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh = ii ) i=(r + (r i j ) i< j = ii ) i=(r + ( ) i< j di d j = i c i ) i=(d + i< j = M where M = i c i ) i=(d + i< j { if vi C where c i = d i d j if v i C d i d j The questio of whe does the graph eergy becomes a ratioal umber was aswered by Bapat ad Pati i their article [4] Similar result for Laplacia miimum coverig Radić eergy is obtaied i the followig theorem Theorem 7 Let G be a graph with a miimum coverig set C If the sum of the absolute values of Laplacia miimum coverig Radić eigevalues is a ratioal umber the ρ i C (mod) i= Proof Let ρ ρ ρ be Laplacia miimum coverig Radić eigevalues of a graph G of which ρ ρ ρ r are positive ad the rest are o-positive the ρ i = (ρ + ρ + + ρ r ) (ρ r+ + + ρ ) = (ρ + ρ + + ρ r ) (ρ + ρ + + ρ ) i= = (ρ + ρ + + ρ r ) ρ i = (ρ + ρ + + ρ r ) ( E C ) i= = (ρ + ρ + + ρ r E ) + C Therefore ρ i C (mod) () i= Theorem 8 Let G be a graph with vertices m edges ad C is a miimum coverig set of G If the sum of the absolute Laplacia miimum coverig Radić eigevalues is a ratioal umber the LRE C (G) ( C + t m C + t + m) where t is a iteger such that ρ i C (mod) i= Proof We kow that ρ i m ρ i + m i= i= Commuicatios i Mathematics ad Applicatios Vol 9 No pp

15 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 8 ie LRE C (G) ρ i + m = C + t + m (from ()) i= Also LRE C (G) ρ i m = C + t m (from ()) i= ie LRE C (G) ( C + t m C + t + m) The above result is similar to Parity Theorem 7 of [] Bouds for Laplacia Miimum Coverig Radić Eergy MClellad s [] gave upper ad lower bouds for ordiary eergy of a graph Similar bouds for LRE C (G) are give i the followig theorem Theorem 9 (Upper boud) Let G be a graph with vertices m edges ad C is a miimum coverig set of a graph G The LRE C (G) M + m Proof Cauchy-Schwarz iequality is ( ) ( )( ) a i b i a i b i i= i= i= Put a i = b i = ρ i the ( ( )( ) ρ i ) ρ i i= i= i= ie ( ρ i ) M i= Therefore ρ i M i= By Triagle iequality ρ i m ρ i + m i = ie ρ i m ρ i + m i Thus ρ i m ρ i + i= i= i= Therefore LRE C (G) M + m m M + m Commuicatios i Mathematics ad Applicatios Vol 9 No pp

16 8 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh Theorem (Upper boud) Let G be a graph with vertices m edges ad C is a miimum coverig set of G The LRE C (G) M + 4m( C m) Proof Usig Cauchy-Schwarz iequality ( ) ( )( ) a i b i a i b i i= i= i= Put a i = b i = ρi m the ( ρ i m ( )( ) ρ i m ie i= i= i= [ [LRE C (G)] ρ i + 4m i= i= 4m = [M + 4m 8m Therefore LRE C (G) M + 4m( C m) ) i= + 4m C ] ρ i = [M + 4m 4m ] = M + 4m( C m) ] (m C ) Theorem (Lower boud) Let G be a graph with vertices ad m edges ad C is a miimum coverig set of G If D = det LR C (G) the LRE C (G) M + ( )D m Proof Cosider [ ( ) ( ) ρ i ] = ρ i ρ j = i i= i= j= i= ρ + ρ i ρ j i j Therefore ( ρ i ρ j = ρ i ) ρ i () i j i= i= Applyig iequality betwee the arithmetic ad geometric meas for ( ) terms we have ρ i ρ j i j [ ] ( ) ρ i ρ j ( ) ie i j [ ρ i ρ j ( ) i j Usig () we get ( ρ i ) i= ρ i ρ j i j [ ρ i ( ) i= ( [ ρ i ) M ( ) i= ρ i i= ] ( ) ( ) ] ρ i ( ) i= ] Commuicatios i Mathematics ad Applicatios Vol 9 No pp

17 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 8 ( [ ] ρ i ) M + ( ) ρ i i= i= Therefore ρ i M + ( )D i= We kow that ρ i m ρ i m i () ie ρ i m ρ i m i m ρ i ρ i m ie ie i= i= i= ρ i m LRE C (G) i= Therefore LRE C (G) LRE C (G) ρ i m M + ( )D m (from ()) i= M + ( )D m Theorem If ρ (G) is the largest miimum coverig Radić eige value of LR C (G) the ρ (G) m C R(G) Proof For ay ozero vector X we have by [] { X } AX ρ (A) = max X X X Therefore m C i < j ρ (G) J AJ J J = where J is a uit colum matrix di d j m C R(G) = Just like Koole ad Moulto s [6] upper boud for eergy of a graph a upper boud for LRE C (G) is give i the followig theorem ( Theorem If G is a graph with vertices ad m edges ad C + ) i< j d i d the j LRE C (G) M + ( ) M C + i< j d i d j Commuicatios i Mathematics ad Applicatios Vol 9 No pp

18 84 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh Proof Cauchy-Schwartz iequality is [ ] ( )( ) a i b i i= a i i= Put a i = b i = ρ i the ( ρ i ) = i= i= b i i= ρ i i= [LRE C (G) ρ ] ( )(M ρ ) LRE C (G) ρ + ( )(M ρ ) Let f (x) = x + ( )(M x ) For decreasig fuctio f x( ) (x) ( ( ) C + M x M Sice (M) we have M ρ Therefore ( ) M f (ρ ) f ie ( ) M LRE C (G) f (ρ ) f ie ( ) M LRE C (G) f ie LRE C (G) M [ ( ) M ] ( + ) M i< j ) d i d x j Milovaović [4] bouds for Laplacia miimum coverig Radić eergy of a graph are give i the followig theorem Theorem 4 Let G be a graph with vertices ad m edges Let ρ ρ ρ be a o-icreasig order of eigevalues of LR C (G) If C is miimum coverig set the LRE C (G) M α()( ρ ρ ) m where α()= [ ]( [ ]) [x] deotes greatest iteger part of real umber ad M = i c i ) i=(d + i< j d i d j Commuicatios i Mathematics ad Applicatios Vol 9 No pp

19 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 85 where c i = { if vi C if v i C Proof Let a a a a A ad b b b b B be real umbers such that a a i A ad b b i B i = the the followig iequality is valid a i b i a i b i α()(a a)(b b) i= i= i= If a i = ρ i b i = ρ i a = b = ρ ad A = b = ρ ( ρ i ρ i ) α()( ρ ρ ) i= i= But ( ρ i ) M i= ( ) M ρ i α()( ρ ρ ) i= Sice Hece ( ρ i i= LRE C (G) = ) i= M α()( ρ ρ ) ρ i m ( ) ρ i m i= LRE C (G) M α()( ρ ρ ) m Theorem 5 Let G be a graph with vertices ad m edges Let ρ ρ ρ > be a o-icreasig order of eigevalues of LR C (G) ad C is miimum coverig set the LRE C (G) M + ρ ρ m ( ρ + ρ ) where M = (d i c i ) + i= i< j d i d j Here c i = { if vi C if v i C Proof Let a i b i r ad R be real umbers satisfyig ra i b i Ra i the we have the followig iequality b i + rr a i (r + R) a i b i i= i= i= i= i= Put b i = ρ i a i = r = ρ ad R = ρ ρ i + ρ ρ ( ρ + ρ ) ρ i ie M + ρ ρ ( ρ + ρ ) i= ρ i i= Commuicatios i Mathematics ad Applicatios Vol 9 No pp

20 86 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh We kow that Therefore ρ i M + ρ ρ ( ρ + ρ ) i= LRE C (G) = ρ i m i= LRE C (G) ρ i m i= LRE C (G) M + ρ ρ ( ρ + ρ ) m Coclusios It was proved i this article that the miimum coverig Radić eergy of a graph G depeds o the coverig set that we take for cosideratio Upper ad lower bouds for miimum coverig Radić eergy are established A geeralized expressio for miimum coverig Radić eergies for star graph complete graph thor graph of complete graph crow graph complete bipartite graph cocktail party graph ad friedship graphs are also computed Competig Iterests The authors declare that they have o competig iterests Authors Cotributios All the authors cotributed sigificatly i writig this article The authors read ad approved the fial mauscript Refereces [] C Adiga A Bayad I Gutma ad SA Sriivas The miimum coverig eergy of a graph Kragujevac J Sci 4 () 9 56 [] T Aleksić Upper bouds for Laplacia eergy of graphs MATCH Commu Math Comput Chem 6 (8) [] RB Bapat Graphs ad Matrices page Hidusta Book Agecy () [4] RB Bapat ad S Pati Eergy of a graph is ever a odd iteger Bull Kerala Math Assoc () 9 [5] M Bieracki H Pidek ad C Ryll-Nadzewski Sur ue iequalite etre des iegrales defies Aales Uiv Marie Curie-Sklodowska A4 (95) 4 [6] SB Bozkurt AD Gügör ad I Gutma Radić spectral radius ad Radić eergy Commu Math Comput Chem 64() 9 5 [7] SB Bozkurt ad D Bozkurt Sharp upper bouds for eergy ad Radić eergy Commu Math Comput Chem 7 () [8] V Cosoi ad R Todeschii New spectral idex for molecule descriptio MATCH Commu Math Comput Chem 6 (8) 4 Commuicatios i Mathematics ad Applicatios Vol 9 No pp

21 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh 87 [9] D Cvetković ad I Gutma (eds) Applicatios of Graph Spectra Mathematical Istitutio Belgrade (9) [] D Cvetković ad I Gutma (eds) Selected Topics o Applicatios of Graph Spectra Mathematical Istitute Belgrade () [] NNM de Abreu CTM Viagre AS Boifácio ad I Gutma The Laplacia eergy of some Laplacia itegral graphs MATCH Commu Math Comput Chem 6 (8) [] A Dilek Made New bouds o the icidece eergy Radić eergy ad Radić Estrada idex Commu Math Comput Chem 74 (5) [] JB Diaz ad FT Matcalf Stroger forms of a class iequalities of G Polja-G Szego ad LV Katorovich Bull Amer Math Soc 69 (96) [4] GH Fath-Tabar AR Ashrafi ad I Gutma Note o Laplacia eergy of graphs Bull Acad Serbe Sci Arts (Cl Math Natur) 7 (8) [5] A Graovac I Gutma ad N Triajstić Topological Approach to the Chemistry of Cojugated Molecules Spriger Berli (977) [6] I Gutma The eergy of a graph Ber Math-Statist Sekt Forschugsz Graz (978) [7] I Gutma The eergy of a graph: old ad ew results edited by A Bette A Kohert R Laue ad A Wasserma Algebraic Combiatorics ad Applicatios Spriger Berli () pp 96 [8] I Gutma NMM de Abreu CTM Viagre AS Boifácio ad S Radeković Relatio betwee eergy ad Laplacia eergy MATCH Commu Math Comput Chem 59 (8) 4 54 [9] I Gutma ad OE Polasky Mathematical Cocepts i Orgaic Chemistry Spriger Berli (986) [] I Gutma X Li ad J Zhag i Graph Eergy edited by M Dehmer ad F Emmert-Streib Aalysis of Complex Networks from Biology to Liguistics Wiley-VCH Weiheim (9) pp [] I Gutma ad B Zhou Laplacia eergy of a graph Li Algebra Appl 44 (6) 9 7 [] H Liu M Lu ad F Tia Some upper bouds for the eergy of graphs Joural of Mathematical Chemistry 4() (7) [] BJ McClellad Properties of the latet roots of a matrix: The estimatio of π-electro eergies J Chem Phys 54 (97) [4] I Ž Milovaović EI Milovaović ad A Zakić A short ote o graph eergy Commu Math Comput Chem 7 (4) 79 8 [5] KC Das S Sorgu ad I Gutma O Radić eergy Commu Math Comput Chem 7 (5) 8 9 [6] JH Koole ad V Moulto Maximal eergy graphs Adv Appl Math 6 () 47 5 [7] J Liu ad B Liu O relatio betwee eergy ad Laplacia eergy MATCH Commu Math Comput Chem 6 (9) 4 46 [8] MR Rajesh Kaa ad R Jagadeesh Miimum coverig Radic eergy of a graph Advaces i Liear Algebra ad Matrix Theory 6 (6) 6 [9] M Radić O characterizatio of molecular brachig J Amer Chem Soc 97 (975) [] S Radeković ad I Gutma Total π-electro eergy ad Laplacia eergy: How far the aalogy goes? J Serb Chem Soc 7 (7) 4 5 Commuicatios i Mathematics ad Applicatios Vol 9 No pp

22 88 Laplacia Miimum coverig Radić Eergy of A Graph: MR Rajesh Kaa ad R Jagadeesh [] M Robbiao ad R Jiméez Applicatios of a theorem by Ky Fa i the theory of Laplacia eergy of graphs MATCH Commu Math Comput Chem 6 (9) [] D Stevaović I Staković ad M Milošević More o the relatio betwee eergy ad Laplacia eergy of graphs MATCH Commu Math Comput Chem 6 (9) 95 4 [] H Wag ad H Hua Note o Laplacia eergy of graphs MATCH Commu Math Comput Chem 59 (8) 7 8 [4] B Zhou O the sum of powers of the Laplacia eigevalues of graphs Li Algebra Appl 49 (8) 9 46 [5] B Zhou New upper bouds for Laplacia eergy MATCH Commu Math Comput Chem 6 (9) [6] B Zhou I Gutma ad T Aleksić A ote o Laplacia eergy of graphs MATCH Commu Math Comput Chem 6 (8) [7] B Zhou ad I Gutma O Laplacia eergy of graphs MATCH Commu Math Comput Chem 57 (7) [8] B Zhou ad I Gutma Nordhaus-Gaddum-type relatios for the eergy ad Laplacia eergy of graphs Bull Acad Serbe Sci Arts (Cl Math Natur) 4 (7) Commuicatios i Mathematics ad Applicatios Vol 9 No pp