Turkish Joural of Aalysis ad Number Theory, 7, Vol, No 6, -9 Available olie at http://pubssciepubcom/tat//6/ Sciece ad Educatio Publishig DOI:69/tat--6- Symmetric Divisio Deg Eergy of a Graph K N Prakasha, P Siva Kota Reddy, Ismail Naci Cagul 3,* Mathematics, Vidyavardhaka College of Egieerig, Mysuru, Idia Mathematics, Siddagaga Istitute of Techology, Tumkur, Idia 3 Mathematics, Uludag Uiversity, Bursa, Turkey *Correspodig author: cagul@gmailcom Received July 8, 7; Revised August 9, 7; Accepted September, 7 Abstract The purpose of this paper is to itroduce ad ivestigate the symmetric divisio deg eergy SDDE(G of a graph We establish upper ad lower bouds for SDDE(G Also the symmetric divisio deg eergy for certai graphs with oe edge deleted are calculated Keywords: symmetric divisio deg idex, symmetric divisio deg eigevalues, symmetric divisio deg eergy, k-complemet, k(i-complemet, edge deletio Cite This Article: K N Prakasha, P Siva Kota Reddy, ad Ismail Naci Cagul, Symmetric Divisio Deg Eergy of a Graph Turkish Joural of Aalysis ad Number Theory, vol, o 6 (7: -9 doi: 69/tat--6- Itroductio v v v be the Let G be a simple graph ad let {,,, } set of its vertices Let i, {,,, } ad If two vertices v i v of G are adacet, the we use the otatio vi ~ v For a vertex vi V( G, the degree of v i will be dv or briefly by d deoted by i i I mathematical chemistry, topological idices play a importat role due to their coutless applicatios There are may topological idices such as Radić idex, sum-coectivity idex, atom bod coectivity idex, Zagreb idices, etc Oe of those umerical descriptors, the symmetric divisio deg idex, is icluded i the list of 48 discrete Adriatic idices ad is a very good predictor of total surface area of polychlorobipheyls (PCB The symmetric divisio deg idex of a graph G is defied by mi, max, R= RG = + max, mi, i i i~ i i The cocept of the symmetric divisio deg idex motivates oe to associate a symmetric square matrix SDD(G to a graph G The symmetric divisio deg matrix SDD G = S is, by this reaso, defied as ( i { } {, } mi i, max i, + Si = max di, d mi di d if vi ~ v, otherwise The Symmetric Divisio Deg Eergy of a Graph Let G be a simple, fiite, udirected graph The classical eergy E(G is defied as the sum of the absolute values of the eigevalues of its adacecy matrix For more details o eergy of a graph, see [,3] Let SDD(G be the symmetric divisio deg matrix The characteristic polyomial of SDD(G will be deoted by φ SDD ( G, λ ad defied as = ( φ SDD G, λ det λi SDD ( G Sice the symmetric divisio deg matrix is real ad symmetric, its eigevalues are real umbers ad we label them i o-icreasig order λ λ λ The symmetric divisio deg eergy of G is similarly defied by SDDE( G λi i= = ( This paper is orgaized as follows I Sectio 3, we give some basic properties of symmetric divisio deg eergy of a graph I Sectio 4, symmetric divisio deg eergy of some specific graphs are obtaied I Sectio, we fid symmetric divisio deg eergy of some complemets of some specific graphs I Sectio 6, the symmetric divisio deg eergy for certai graphs with oe edge deleted are calculated ad fially i Sectio 7, some ope problems are give 3 Some Basic Properties of Symmetric Divisio Deg Eergy of a Graph Let us defie the umber P as
Turkish Joural of Aalysis ad Number Theory 3 di + d P = i< di d The we have Propositio 3 The first three coefficiets of the polyomial φ SDD ( G, λ are as follows: (i a =, (ii a =, (iii a = P Proof (i By the defiitio of the polyomial = [ ] φ SDD G, λ det λi SDD ( G, we get a = (ii The sum of determiats of all pricipal SDD G submatrices of SDD( G is equal to the trace of implyig that tr( SDD( G a = = (iii By the defiitio, we have aii = a < i i a = aiia aiai < i = P ai a Propositio 3 If λ, λ,, λ are the symmetric divisio deg eigevalues of SDD( G, the Proof It follows as λi = P i= λi = aiai i= i= = ( ai ( aii = + = ( ai i< i= i< = P Usig this result, we ow obtai lower ad upper bouds for the symmetric divisio deg eergy of a graph: Theorem 33 Let G be a graph with vertices The SDDE G P Proof Let λ, λ,, λ be the eigevalues of SDD( G By the Cauchy-Schwartz iequality we have ab i i ai bi i= i= i= Let a =, b = λ The i i i implyig that ad hece we get λi λ i i= i= i= [ SDDE] P [ SDDE] as a upper boud Theorem 34 Let G be a graph with vertices If R = det SDD G, the P + ( SDDE G P R Proof By defiitio, we have ( SDDE( G = λ i i= = λ i i= = λ = + λi λi λ i= i Usig arithmetic-geometric mea iequality, we have Therefore, λi λ λi λ ( i i ( SDDE( G λi + ( λi λ i= i Thus, ( ( λi + ( λi i= i = λi + R i= = P+ R + ( SDDE G P R Let λ ad λ are the miimum ad maximum values of all λ i s The the followig results ca easily be prove by meas of the above results: Theorem 3 For a graph G of order, 4 ( SDDE G P λ λ
4 Turkish Joural of Aalysis ad Number Theory Theorem 36 For a graph G of order with o-zero eigevalues, we have λλ P SDDE ( G ( λ + λ Theorem 37 Let G be a graph of order Let λ λ λ3 λ be the eigevalues i icreasig order The SDDE G λ λ λ + P + λ 4 Symmetric Divisio Deg Eergy of Some Graph Types I this sectio, we calculate the symmetric divisio deg eergy of some well-kow ad frequetly used graph types icludig complete, cycle, star, friedship, cocktail party, double star, Dutch widmill, crow ad complete bipartite graphs Theorem 4 The symmetric divisio deg eergy of a complete graph K is SDDE K = 4 4 Proof Let K be the complete graph with vertex set V = { v, v,, v } For this graph, the symmetric divisio deg matrix is SDD K = The characteristic equatio the becomes ( λ ( λ ( ad the spectrum would be SpecSDD + = ( K Therefore, SDDE ( K = 4 4 = Theorem 4 The symmetric divisio deg eergy of the cycle graph C is SDDE C π m = 8 + 4cos m=, m Proof The symmetric divisio deg matrix correspodig to the cycle graph C is SDD C = This is a circullat matrix of order Its eigevalues are 4, for m= λm = -4, for m= π m 4cos, for < m <, < m Therefore the symmetric divisio deg eergy is SDD C π m = 4 + 4 + 4cos m=, m Theorem 43 The symmetric divisio deg eergy of the star graph K, is SDDE K (, + = Proof Let K, be the star graph with vertex set { } V = v, v,, v with v deotes the cetral vertex The symmetric divisio deg matrix is SDD K, = The characteristic equatio becomes λ + + + + + ( + λ = ad therefore, the spectrum would have a + a ad times Therefore, +,
Turkish Joural of Aalysis ad Number Theory SDDE K (, + = 3 Defiitio 44 The friedship graph, deoted by F, is defied as the graph obtaied by takig copies of the cycle graph C 3 with a vertex i commo 3 It is clear that V ( F = + Theorem 4 The symmetric divisio deg eergy of the 3 friedship graph F is ( 4 3 8 + 6 + 4+ 8 SDDE F = 4 + 3 Proof Let F be the friedship graph with + vertices ad let v be the commo vertex The symmetric divisio deg matrix is The characteristic equatio becomes + + + + + + + + + + + + ( + λ λ+ λ λ = implyig that the spectrum has times, times, 4 + 4 + + a + ad a Therefore, we get ( 4 + 4 + + 4 3 8 + 6 + 4+ 8 SDDE F = 4 + Theorem 46 The symmetric divisio deg eergy of the cocktail party graph K is SDDE K = 8 8 Proof Let K be the cocktail party graph of order havig vertex set { u, u,, u, v, v,, v } The symmetric divisio deg matrix is SDD K = I that case, the characteristic equatio is ( ( λ λ+ 4 λ 4 4 = ad hece the spectrum becomes Spec SDD ( K 4 4 4 = Therefore we arrive at the required result: SDDE K = 8 8 Theorem 47 The symmetric divisio deg eergy of the double star graph S, is ( ( + SDDE S, = 4+ 4 Proof The symmetric divisio deg matrix is + + + + + + + + Hece, the spectrum would have 4 times,
6 Turkish Joural of Aalysis ad Number Theory ( ( + a + +, ad Therefore, we get ( ( + a + ( ( + SDDE S, = 4 + Defiitio 48 A graph obtaied by oiig copies of the cycle graph C 4 of legth 4 at a commo vertex is called a Dutch widmill graph It will be deoted by D 4 It is clear that the Dutch widmill graph D 4 has 3 + vertices ad 4 edges Theorem 49 The symmetric divisio deg eergy of the Dutch widmill graph D 4 is SDDE D4 = 8 + + 3 Proof Recall that D 4 has 3 + vertices The the symmetric divisio deg matrix is + + + + + + + Hece the characteristic equatio will be + + 6 8 + 8 = λ λ λ λ ad therefore the spectrum would have times 8, times 8, + times, + 6 Therefore, it is directly see that SDDE D4 = 8 + + 3 + 6 ad Theorem 4 The symmetric divisio deg eergy of crow graph S is SDDE S ( = 8 8 Proof Let S be the crow graph of order ad let the u, u,, u, v, v,, v vertex set of this graph be { } The symmetric divisio deg matrix of S is Therefore the characteristic equatio is ( λ ( λ ( λ ( ( λ ( + + = implyig that the spectrum has a, a, times ad times Therefore we obtai SDDE S ( = 8 8 Theorem 4 The symmetric divisio deg eergy of the complete bipartite graph K m, of order m with vertex set {,,,,,, } u u v v is m m + SDDE ( Km, = m Proof The symmetric divisio deg matrix of the complete bipartite graph K m, is m + m + m m m + m + m m m + m + m m m + m + m m m + m + m m m + m + m m The the characteristic equatio is + + + m m m λ λ λ+ = m m ad therefore the spectrum has a m +, m m+
Turkish Joural of Aalysis ad Number Theory 7 m + times ad a Therefore, we get m m + SDDE ( Km, = m Symmetric Divisio Deg Eergy of Complemets Defiitio [] Let G be a graph ad Pk = { V, V,, Vk} be a partitio of its vertex set V The the k-complemet of G is deoted by ( G k ad obtaied as follows: For all V i ad V i P k, i, remove the edges betwee V i ad V ad add the edges betwee the vertices of V i ad V which are ot i G Defiitio [] Let G be a graph ad Pk = { V, V,, Vk} be a partitio of its vertex set V The the k( i -complemet of G is deoted by ki G ad obtaied as follows: For each set V r i P k, remove the edges of G oiig the vertices withi V r ad add the edges of G (complemet of G oiig the vertices of V r There is usually a ice relatio betwee some properties of a graph ad its complemet Here we ivestigate the relatio betwee some special graph classes ad their complemets i terms of the symmetric divisio deg eergy Theorem 3 The symmetric divisio deg eergy of the complemet K of the complete graph K is SDDE ( K = Proof Let K be the complete graph with vertex set V = { v, v,, v } The symmetric divisio deg coectivity matrix of the complemet of the complete graph K is SDD K = Clearly, the characteristic equatio is λ = implyig SDDE ( K = Theorem 4 The symmetric divisio deg eergy of the complemet K, of the star graph K, is ( SDDE K, = 4 8 Proof Let K, be the complemet of the star graph V = v, v,, v where v is the cetral vertex The symmetric divisio deg matrix is with vertex set { } SDD K (, = The correspodig characteristic equatio is ( ( ( λ λ λ ad therefore the spectrum is SpecSDD Therefore, 4 + = ( K, 4 = ( SDDE K, = 4 8 Theorem The symmetric divisio deg eergy of the complemet K of the cocktail party graph K of order is ( SDDE K = 4 Proof Let K be the cocktail party graph of order havig the vertex set {,,,, v, v,, v } The correspodig symmetric divisio deg matrix is ad the characteristic equatio becomes ( λ ( λ + = implyig that the spectrum would be Therefore, SpecSDD ( K ( = SDDE K = 4
8 Turkish Joural of Aalysis ad Number Theory The characteristic polyomial is ( ( λ λ+ 4 λ 4 = ad therefore the symmetric divisio deg spectrum has times 4, times, a 4 ad a implyig that the symmetric divisio deg eergy is SDDE K = 8 8 ( Figure Double star graph with its (i-complemet Theorem 6 The symmetric divisio deg eergy of (i-complemet of double star graph S, is (, ( 4 SDDE S = + A + B i where A= ad B = 6 3+ Proof The symmetric divisio deg matrix for (i-complemet of double star graph is Therefore the spectrum has 4 times, a + A + A 4 6+ B 4 6+ B, a, a ad a Therefore we obtai the required result Theorem 7 The symmetric divisio deg eergy of -complemet of cocktail party graph K is SDDE K 8 ( = ( Proof Cosider the -complemet of the cocktail party graph K The symmetric divisio deg matrix is ( 6 Symmetric Divisio Deg Eergy of Graphs with Oe Edge Deleted Edge deletio is very importat i combiatorial calculatios with graphs I this sectio, we obtai the symmetric divisio deg eergy for certai graphs with oe edge deleted This ca be used recursively to calculate the symmetric divisio deg eergy of a give graph Theorem 6 Let e be a edge of the complete graph K The SDDE ( K e is equal to ( 3 ( 3+ + ( 4( 6+ 6+ 3+ Proof The symmetric divisio deg matrix for K e is 6+ 3+ 6+ 3+ 6+ 6+ 3 3 + + 6+ 6+ 3+ 3+ 6+ 6+ 3+ 3+ Therefore the spectrum would have a 3+ 3 3+ + 4 6+ 3+ 3+ 3 3+ + 4 6+ a 3+ 3 times ad a, implyig the result Theorem 6 Let e be a edge of the complete bipartite graph K, The symmetric divisio deg eergy of K, e is equal to 4 3 4 + 4 + 8 6+ 4 (,
Turkish Joural of Aalysis ad Number Theory 9 Proof The symmetric divisio deg matrix for K, e is + ( + ( + ( + ( + + A Hece, the spectrum would have a, a + A + A A, a, a, ad 4 times implyig the result The followig result ca easily be prove as above: Lemma 63 Let K, be the star graph with vertices ad let e be a edge of it The SDDE ( K, e = SDDE ( K, for 3 7 Some Ope Problems Ope problem 7 With respect to symmetric divisio deg, determie the class of graphs which are co-spectral ad characterize them Ope problem 7 With respect to symmetric divisio deg, determie the class of graphs which are hypereergetic ad characterize them Ope problem 73 With respect to symmetric divisio deg, determie the class of graphs whose symmetric divisio deg eergy ad symmetric divisio deg eergy of their complemets are equal Ope problem 74 With respect to symmetric divisio deg, determie the class of o-co-spectral graphs which are equieergetic Ope problem 7 Determie the class of graphs whose symmetric divisio deg eergy is equal to usual eergy Refereces [] Alexader, V, Upper ad lower bouds of symmetric divisio deg idex, Iraia Joural of Mathematical Chemistry, ( (4, 9-98 [] Gutma, I, The eergy of a graph, Ber Math Stat Sekt Forschugsz Graz, 3 (978, - [3] Gutma, I, The eergy of a graph: old ad ew results, Combiatorics ad applicatios, A Bette, A Khoer, R Laue ad A Wasserma, (Eds, Spriger, Berli, (, 96- [4] Radić, M, O characterizatio of molecular brachig, J Am Chem Soc, 97 (97, 669-66 [] Sampathkumar, E, Pushpalatha, L, Vekatachalam, C V ad Bhat, P, Geeralized complemets of a graph, Idia J Pure Appl Math, 9(6 (998, 6-639 [6] Todeschii, R, Cosoi, V, Hadbook of Molecular Descriptors, Wiley-VCH, Weiheim, (, 84-9 [7] Todeschii, R, Cosoi, V, Molecular Descriptors for Chemoiformatics, Wiley-VCH, Weiheim, (9, 6-7 [8] Zhou, B, Triastic, N, O Sum-Coectivity Matrix ad Sum-Coectivity Eergy of (Molecular Graphs, Acta Chim Slov, 7 (, 8-3 [9] Zhou, B, Triastic, N, O a ovel coectivity idex, J Math Chem, 46 (9, -7