LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH

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1 Kragujevac Joural of Mathematics Volume 4 018, Pages LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH H J GOWTHAM 1, SABITHA D SOUZA 1, AND PRADEEP G BHAT 1 Abstract Let P = {V 1, V, V 3,, V k } be a partitio of vertex set V G of order k For all V i ad V j i P, i j, remove the edges betwee V i ad V j i graph G ad add the edges betwee V i ad V j which are ot i G The graph G P k thus obtaied is called the k complemet of graph G with respect to a partitio P For each set V r i P, remove the edges of graph G iside V r ad add the edges of G the complemet of G joiig the vertices of V r The graph G P ki thus obtaied is called the ki complemet of graph G with respect to a partitio P I this paper, we study Laplacia eergy of geeralized complemets of some families of graph A effort is made to throw some light o showig variatio i Laplacia eergy due to chages i a partitio of the graph 1 Itroductio Let G be a graph o vertices ad m edges The complemet of a graph G, deoted as G has the same vertex set as that of G, but two vertices are adjacet i G if ad oly if they are ot adjacet i G If G is isomorphic to G the G is said to be self-complemetary graph For all otatios ad termiologies we refer [, 15, 1] E Sampathkumar et al have itroduced two types of geeralized complemets [19] of a graph For completeess we produce these here Let P = {V 1, V, V 3,, V k } be a partitio of vertex set V G of order k For all V i ad V j i P, i j, remove the edges betwee V i ad V j i graph G ad add the edges betwee V i ad V j which are ot i G The graph G P k thus obtaied is called the k complemet of graph G with respect to a partitio P For each set V r i P, remove the edges of graph G iside V r ad add the edges of G joiig the vertices Key words ad phrases Geeralized complemets, Laplacia spectrum, Laplacia eergy 010 Mathematics Subject Classificatio Primary: 05C15 Secodary: 05C50 Received: April 09, 016 Accepted: March 03,

2 300 H J GOWTHAM, S D SOUZA, AND P G BHAT of V r The graph G P ki thus obtaied is called the ki-complemet of graph G with respect to a partitio P The eergy of the graph is first defied by Iva Gutma [10] i 1978 as the sum of absolute eigevalues of graph G For more iformatio o eergy of a graph we refer [1,3,5,7 9,1,13,17,18,0] Let DG = diagd 1, d,, d be the diagoal matrix of vertex degrees, ad AG is the adjacecy matrix The LG = DG AG is the Laplacia matrix of graph G The characteristic polyomial of the Laplacia matrix is deoted by φlg, µ = detµi LG Let {µ 1, µ,, µ } be the Laplacia eigevalues of graph G, ie, the roots of φlg, µ The Laplacia eergy[14], deoted by LEG, is defied as LEG = µ i m The Laplacia eergy LEG is a very recetly defied graph ivariat The basic properties for Laplacia eergy have bee established i [4,6,11,14,,3], ad it has foud remarkable chemical applicatios I this paper we study the Laplacia eergy of geeralized complemets of some classes of graphs Prelimiaries Propositio 1 [19] The k-complemet ad ki complemet of G are related as follows: i G P k = G P ki ; ii G P ki = G P k Theorem 1 [6] Let G be a graph with vertices ad G be its complemet If the Laplacia spectrum of G is {µ 1, µ,, µ }, the the Laplacia spectrum of G is { µ 1, µ,, µ 1, 0} Defiitio 1 [6] Let f i, i = 1,,, k, 1 k p, be idepedet edges of the complete graph K p, p 3 The graph Kb p k is obtaied by deletig f i, i = 1,,, k, from K p I additio Kb p 0 = K p Propositio [6] For p 3 ad 0 k p, LEKb p k = p + 4 k 4k p p Lemma 1 [16] Let A = [ ] A0 A 1 A 1 A 0 be a block symmetric matrix The the eigevalues of A are the eigevalues of the matrices A 0 + A 1 ad A 0 A 1

3 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH Laplacia eergy of geeralized complemets of classes of graphs Now we fid Laplacia eergy of geeralized complemets of some stadard graphs like complete, complete bipartite, path, cycle, double star, friedship ad cocktail party graph For some graphs we take partitio of order k ad for some partitio of order two Theorem 31 Let P = {V 1, V,, V k } be a partitio of the complete graph K i If < V i >= K i, for i = 1,,,, the k k i C k k k i C K P k = + i 1 i ad C k i C K P ki =k 1 + k + k C k i C + i 1 i ii If V i = ad oe of V i = 1 whe is odd, the LE K P k = 4k k ad K P 3 kk 1 ki = Proof For a partitio P = {V 1, V,, V k }, let G = K P k be a graph i If < V i >= K i, for i = 1,,,, the G is the uio of k discoected complete subgraphs of order i such that i = If i, the Laplacia spectrum of K i cosists of {0, i i 1 times}, { i = 1,,, } 0 The the Laplacia spectrum of K P 1 k is k k k 1 k ic ad the average degree of K P k =, k k i C k k k i C K P k = + i 1 i, where V i = i, i = 1,,, k

4 30 H J GOWTHAM, S D SOUZA, AND P G BHAT By otig that K P ki = K P k ad from Theorem 1, we obtai Lapla- } { 0 cia spectrum of K P ki is 1 k 1 k k 1 k C ic average degree of K P ki is, K P ki =k 1 + k + C k i C k C k i C + i 1 i, ad the where V i = i, i = 1,,, k, if is eve, ii If V i =, the k = + 1, if is odd It follows by substitutig the value of k i Theorem 31 of statemet i Also, the Laplacia spectrum of G is give by {0k times, k times} Average degree of G is m = k Thus, LEG = µ i m k k = k + k k k k k = + 4k k =, k 1 Hece, Laplacia spectrum of G cosists of {0, k times, k 1 times} Average degree of G is m = 1 k LEG = µ i m 1 k = 1 k + k

5 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH k + k 1 3 kk 1 =, k 1 Remark 31 Let P = {V 1, V,, V k } be a partitio of the complete graph K with V i = ad oe of V i = 1, if is odd The, LE K P k + LE K P ki = 3 4, if is eve, 3 1, if is odd Theorem 3 Let P = {V 1, V,, V k } be a partitio of path P i If ay oe of the pedat vertex is i V 1 or V k, ad remaiig k 1 sets are K s, the P P ki = {kk 1 + k k + 1}, whereas [ ] P P k = LE Kb = + 4 4, for odd 3 ii If ay oe of the o pedat vertex is i V i, 3 i ad remaiig k 1 sets are K s, the P P ki = [ k + kk + ], whereas P P k = [ k + kk 1], k = for odd 5 iii If < V i >= K, the P P 4 ki = k 1 k + 1 ad P P k = 1, for eve Proof i If ay oe of the pedat vertex is i V 1 or V k, ad remaiig k 1 sets are K s, the P P ki is the uio of k 1 K s ad oe isolated vertex Laplacia spectrum of P P ki is {0k times, k times} The average degree of P P ki is k 1, ie, P P ki =k k 1 + k k 1 = [ k + kk + ]

6 304 H J GOWTHAM, S D SOUZA, AND P G BHAT Note that P P k is the graph obtaied from K by deletig all the idepedet edges, P P k = Kb Hece, from Propositio, LE P P k [ ] = LE Kb = ii If ay oe of the o pedat vertex is i V i, 3 i ad remaiig k 1 sets are K s, the P P ki is the uio of K 1,, k K s ad two isolated vertices Hece, Laplacia spectrum of P P ki is {0k times, 1, k times, 3} The average degree of P P k 1 ki is, ie, P P ki =k k 1 + k k k k 1 = [ k + kk + ] Also from Theorem 1, Laplacia spectrum of P P k is {0, 3, k times, 1, k 1 times} Average degree of P P k is 1 k 1 Hece, P P 1 k 1 k = 1 k k 1 + k 1 k k 1 + k 1 = [ k + kk + ] iii If < V i >= K, the P P ki is the uio of k 1 K s ad two isolated vertices Its Laplacia spectrum is {0 k + 1 times, k 1 times} ad average degree is k 1 Hece, P P ki = k + 1 k 1 + k 1 k 1 = 4 k 1 k + 1

7 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH 305 Also from Theorem 1, Laplacia spectrum of P P k is {0, k 1 times, k times} Average degree of P P k is 1 k 1 Thus, P P 1 k = [ + k 1 + k 1 + k 1 k 1] Note that k = for path of eve order, hece, we obtai, P P k = 1 Remark 3 As k = for path of eve order, P P ki = 4 Theorem 33 Let P = {V 1, V,, V k } be a partitio of cycle C i If ay of the V i is K 1 ad remaiig V j s are all K s, where i j The C P [ ki = k + k ], whereas C P 4k k = [ ], for odd 3 ii If each V i cosists of K, the C P ki = [k + k ], whereas C P k = + 4 4, for eve 4 Proof i If ay of the V i is K 1 ad remaiig V j s are all K s, where i j The C P ki is the uio of K 1, ad k 1 K s Hece, Laplacia spectrum of C P ki is {0k 1 times, 1, k 1 times, 3} The average degree of C P ki is k ad LE C ki =k 1 k + 1 k + k 1 k + 3 k = [ k + k ] Also accordig to Theorem 1, Laplacia spectrum of C P k is {0, 3, k 1 times, 1, k times} The average degree of C P k is 1 k Thus LE C P 1 k k = + 1 k 3 1 k + 1

8 306 H J GOWTHAM, S D SOUZA, AND P G BHAT 1 k + k 1 1 k + k = 4k [ ] ii If each V i cosists of K, the C P ki has k compoets of K Hece, Laplacia specrtum of C P ki is {0k times, k times} The average degree of C P ki is k ad LEC P ki = k k + k k = [ k + k ] Also from Theorem 1, Laplacia spectrum of C P k is {0, k times, k 1 times} As C P ki has k edges, C P k has C k edges Also, ote that C P k has K k idepedet edges For cycle of eve order, k =, we have C P k = LE Kb = Theorem 34 Let K m, = {U m, U } be complete bipartite graph with partitio P = {V 1, V } The, i If < V 1 >= K s1,s ad < V >= K m s1, s, where s 1, s deote umber of vertices of V 1 such that s 1 vertices belog to U m ad s vertices belog to U ad s 1 < m, s <, the where ad LE K m, P =q + s1 + s m s + s 1 s 1 + s m s + s 1 s 1 + s + m s + s 1, s 1 + s, if s 1 + s m s + s 1, q = m s + s 1, if m s + s 1 < s 1 + s LE K m, P i = m + ii If V 1 = m 1 such that all the vertices of V 1 are from first partite set of K m, ad V = + 1, the LE K m, P + = ad LE K m, P i = m +

9 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH 307 Proof i If < V 1 >= K s1,s ad < V >= K m s1, s, the Hece, K m, P = K s1 +s,m s +s 1 LE K m, P =q + s1 + s m s + s 1 s 1 + s m s + s 1 s 1 + s + m s + s 1, where s 1 + s, if s 1 + s m s + s 1, q = m s + s 1, if m s + s 1 < s 1 + s Also LE K m, P i = Km K Hece, LE K m, P i = LEKm + LEK = m + ii If V 1 = m 1 such that all the vertices of V 1 are from first partite set of K m, ad V = + 1, the K m, P = K 1,m+ 1 Hece K m, P = + Also Km, P i = K 1 K m+ 1 Thus, K m, P i = m +, which is stated Theorem 35 Let Sm, be double star graph with partitio P = {V 1, V }, such that the vertices of V 1 ad V are of distace two The m + m + 1, if m >, LE m + Sm, i P m + m + m 1 =, if > m, m + 4m +, if m =, m + ad LE Sm, P 1 1m 1 = m + Proof Let V 1 ad V be the partitio of vertices of Sm, such that the vertices of V 1 ad V are of distace two, ie, V 1 ={v 1, v, v 3,, v m 1, u 1 } ad V ={v m, u, u 3,,

10 308 H J GOWTHAM, S D SOUZA, AND P G BHAT u 1, u } We have LSm, P i = v 1 v v 3 v m 1 v m u 1 u u v 1 m v 1 m v m v m m v m m u m u u Cosider det λi L Sm, P i Step 1: Replace R i by R i R i+1, for i = v 1, v, v 3,, v m, v m ad replace R i by R i R i 1, for i = u, u 1,, u 4, u 3 The det λi L Sm, P i is of the form λ m + 1 m λ + 1 λ + m detd Step : I detd, replace C i by C i C i 1, for i = v, v 3, v 4,, v m 1 ad replace C i by C i C i+1, for i = u 1, u,, u, u 1 The it reduces to a ew determiat λ dete =, m 1 1 λ m λ 1 λ ie, dete= λλ λ + m The Laplacia spectrum of Sm, P ki is {0,, + m times, m + 1m times, + 1 times} ad the average degree of Sm, P i is +1+mm+1 Hece, m+ Sm, P mm + 1 i = m mm m mm m m mm m m + 1 m mm m + If m >, Sm, P m + m + 1 i = m +

11 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH 309 If > m, If m =, Sm, P m + m + m 1 i = m + LE Sm, P i 4m + = m + Also from Theorem 1, Laplacia spectrum of Sm, P is {03 times, m 1 times, 1m times, + m } Average degree of Sm, P is m 1 1 Thus, 4+ LE m 1 1 Sm, P =3 4 + m m m m m m 4 + = 4m + m + Theorem 36 Let F be friedship graph with partitio P = {V 1, V }, such that a partitio V 1 cotais cetral vertex ad remaiig vertices are i a partitio V The F P i = + 1 ad LE F P = Proof Let V 1 ad V be the partitio of vertices of F ad a partitio V 1 cotais oly the cetral vertex ad remaiig vertices are i a partitio V We have v 1 v v 3 v 4 v 1 v v +1 v v +1 v v v v LF P i = v v v v v Cosider det λi LF P i

12 310 H J GOWTHAM, S D SOUZA, AND P G BHAT Step 1: Replace R i by R i R i 1, where i = v +1, v, v 1,, v 3, v The we coclude that det λi LF P i is of the form λ 1 detd Step : I detd, replace C i by C i + C i+1, where i = v, v 1, v,, v 1 We get a ew determiat, let it be dete Step 3: I dete, replace R i by R i + R i 1, where i = v 4, v 6, v 8,, v, v, the the etries below the pricipal diagoal are zeros Hece dete = λλ +1 Thus, det λi LF P i = λλ + 1 λ 1 Therefore, Laplacia spectrum of F P i is {0, 1 times, + 1 times} ad the average degree of F P 4 i is Hece, +1 F P 4 i = = + 1 Also from Theorem 1, Laplacia spectrum of F P Average degree of F P is Thus, +1 is {0 + 1 times, times} LE F P = = Theorem 37 Let F be friedship graph with partitio P = {V 1, V }, such that a partitio V 1 cotais oe triagle ad remaiig vertices are i a partitio V The ad F P i = , if = 3, F P = , if > Proof If V 1 ad V be the partitio of F, such that a partitio V 1 cotais oe triagle ad remaiig vertices are i a partitio V The

13 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH 311 v 1 v v 3 v 4 v 5 v 1 v v +1 v +1 v v v v v v LF P i = v v v v v Cosider det λi LF i P Step 1: Replace R i by R i R i 1, where i = v +1, v, v 1,, v 5 ad replace R v4 by R v4 R v1 The det λi LF i P is of the form λ λ 3 1 detd Step : I detd, replace C i by C i + C i+1, where i = v, v 1, v,, v 4 We get a ew determiat, let it be dete Step 3: I dete, replace R i by R i + R i 1, where i = v, v, v 4,, v 6 ad simplifyig we get, λ 0 0 dete = λ λ λ + 1 Thus, = λ 1 λλ 1 det λi L F P i = λ 3 λ 1 1 λ 3 1 Therefore, the Laplacia spectrum of F P i is {03 times, 1 1 times, 3 1 times} ad the average degree of F P i is 4 +1 Hece, +1 LE 4 F P + 1 i = = Also from Theorem 1, Laplacia spectrum of F P is {0, 1 times, 4 1 times, + 1 times} Average degree of F P is

14 31 H J GOWTHAM, S D SOUZA, AND P G BHAT If = 3, the F P 10 4 = = If > 3, LE F P = + 1 Theorem 38 Let K be cocktail party graph with vertex set V = {v 1, v,, v, u 1, u,, u } ad partitio P = {V 1, V }, such that V 1 cotais v i vertices, ad remaiig u i vertices are i V, where i = 1,, 3,, The LE K P i = LE K P = 4 1 Proof Let K be a cocktail party graph with vertex set V = {v 1, v,, v, u 1, u,, u } ad P = {V 1, V } be a partitio of vertices of K such that V 1 = {v 1,, v } ad V ={u 1, u, u 3,, u } We have [ ] LK P 1I I J i = I J 1I [ ] A0 A It is of the form 1 Hece, from Lemma 1, we get, Laplacia spectrum of A 1 A 0 K P i is {0,, 1 times, 1 times} ad the average degree of K P i is 1 Hece, K P 1 i = =4 1 Also from Theorem 1, Laplacia spectrum of K P is {0,, 1 times, + 1 times} Average degree of K P is Thus, LE K P = = 4 1 Theorem 39 For cocktail graph K with a partitio P = {V 1, V,, V }, such that V i cosists of K i respective partitio, where i = 1,,, The LE K P ki = 10 6 ad LE K P k = 4

15 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH 313 Proof Let P = {V 1, V,, V } be a partitio of cocktail party graph K, such that V i cosists of K i respective partitio, where i = 1,,, The, we have v 1 v v 3 v 1 v u 1 u u v v v v LK P ki = v v u u u u [ ] A0 A 1 It is of the form Hece, from Lemma 1, we get Laplacia spectrum of A 1 A 0 K P ki is {0, 4 1 times, 4 times, 4 4 times} ad the average degree of K P ki is 4 3 Hece, LE K P ki = =10 6 Also from Theorem 1, Laplacia spectrum of K P k 4 times} Average degree of K P k is Thus, is {0 times, times, LE K P k = = 4 Theorem 310 For cocktail graph K +1 with a partitio P = {V 1, V,, V + }, such that V i cosists of K i respective partitio, where i = 1,,, ad V j cosists of K 1 i respective partitio, where j = 1, The LE K +1 P ki = Proof For the give partitio ad LE K +1 P k 8 6 = + 1 v 1 v v 3 v +1 u 1 u u u v v v v LK +1 P ki = v v u u u u

16 314 H J GOWTHAM, S D SOUZA, AND P G BHAT [ ] A0 A It is of the form 1 Hece, from Lemma 1, we get Laplacia Spectrum A 1 A 0 of K +1 P ki is {0, times, 4 times, 4 + times} ad the average degree of K +1 P ki is 44 + Hece, 4+ K +1 P 44 + ki = = + 1 Also from Theorem 1, Laplacia spectrum of K +1 P k is {0 + 1 times, + 1 times, 4 times} Average degree of K +1 P k is 4+1 Thus, K +1 P k = = Refereces [1] R Balakrisha, The eergy of a graph, Liear Algebra Appl , [] R B Bapat, Graphs ad Matrices, Spriger-Hidusta Book Agecy, Lodo, 011 [3] R B Bapat ad S Pati, Eergy of a graph is ever a odd iteger, Bull Kerala Math Assoc 1 004, [4] P G Bhat ad S D Souza, Color sigless laplacia eergy of graphs, AKCE It J Graphs Comb 017, [5] D M Cvetković, M Doob ad H Sachs, Spectra of Graphs, Theory ad Applicatios, third ed, Joha Ambrosius Barth, Heidelberg, 1995 [6] N M M de Abreu, C T M Viagre, A S Boifácio ad I Gutma, The laplacia eergy of some laplacia itegral graphs, MATCH Commu Math Comput Chem , [7] A Graovac, I Gutma ad N Triajstic, Topological Approach to the Chemistry of Cojugated Molecules, Spriger-Verlag, Berli, 1977 [8] I Gutma, The eergy of a graph, Ber Math Stat Sekt Forschugsz Graz , 1 [9] I Gutma, Topological studies o hetero cojugated molecules, Z Naturforch , [10] I Gutma, The eergy of a graph: old ad ew results, i: A Bette, A Kohert, R Laue ad A Wasserma Eds, Algebraic Combiatorics ad Applicatios, Spriger-Verlag, Berli, 001, pp [11] I Gutma, Topology ad stability of cojugated hydrocarbos the depedece of total π-electro eergy o molecular topology, J Serb Chem Soc , [1] I Gutma ad M Mateljević, Topological studies o hetero cojugated molecules, J Math Chem , [13] I Gutma ad O E Polasky, Mathematical Cocepts i Orgaic Chemistry, Spriger-Verlag, Berli, 1986 [14] I Gutma ad B Zhou, Laplacia eergy of a graph, Liear Algebra Appl , 9 37 [15] F Harary, Graph Theory, Narosa Publishig House, New Delhi, 1989

17 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH 315 [16] G Idulal, I Gutma ad A Vijayakumar, O distace eergy of graphs, MATCH Commu Math Comput Chem , [17] X Li, Y Shi ad I Gutma, Graph Eergy, Spriger, New York, 010 [18] S Pirzada ad I Gutma, Eergy of a graph is ever the square root of a odd iteger, Appl Aal Discr Math 008, [19] E Sampathkumar, L Pushpalatha, C V Vekatachalam ad P G Bhat, Geeralized complemets of a graph, Idia J Pure Appl Math , [0] D Stevaović, O spectral radius ad eergy of complete multipartite graphs, Ars Math Cotemp 9 015, [1] D B West, Itroductio to Graph Theory, Pretice Hall, New Delhi, 001 [] B Zhou, More o eergy ad laplacia eergy, MATCH Commu Math Comput Chem , [3] B Zhou, I Gutma ad T Aleksić, A ote o laplacia eergy of graphs, MATCH Commu Math Comput Chem , Departmet of Mathematics, MIT, Maipal Uiversity, Maipal address: gowthamhalgar@gmailcom address: sabithadsouza@maipaledu address: pgbhat@maipaledu