Research Article On the Number of Spanning Trees of Graphs

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1 e Scetfc World Joural, Artcle ID , 5 pages Research Artcle O the Number of Spag Trees of Graphs F Burcu Bozkurt ad DurmuG Bozkurt Departmet of Mathematcs, Scece Faculty, Selçuk Uversty, Alaedd Keykubat Campus, Koya, Turkey Correspodece should be addressed to Ş Burcu Bozkurt; sbbozkurt@selcukedutr Receved 29 August 203; Accepted 24 December 203; Publshed 0 February 204 Academc Edtors: C D Foseca ad A Jaballah Copyrght 204 Ş B Bozkurt ad D Bozkurt Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted We establsh some bouds for the umber of spag trees of coected graphs terms of the umber of vertces,theumber of edges m, maxmum vertex degree Δ, mmum vertex degree δ,frstzagrebdexm, ad RadćdexR Itroducto Let G be a smple coected graph wth vertces ad m edges Let VG = {V, V 2,,V } be the vertex set ad EG = {e,e 2,,e m } theedgesetofgifaytwovertcesv ad V j of G are adjacet, that s, V V j EG,theweusetheotato V V j ForV VG, the degree of the vertex V, deoted by d, s the umber of the vertces adjacet to V LetΔ, Δ 2,ad δ bethemaxmum,thesecodmaxmum,adthemmum vertex degree of G,respectvely Let M =M G = = d2 be the frst Zagreb dex [] ad R α =R α G = V V j d d j α the geeral Radć dex [2] of the graph G, whereα=0s a fxed real umber Note that the Radć dex R =R G = V V j /d d j s also well studed the lterature For more detals o R,see[3, 4] Let K, K p,q p + q =, ads deote the complete graph, the complete bpartte graph, ad the star graph of order, respectvelyletg ebe the graph obtaed by deletg the edge e from the graph G ad let G be the complemet of G LetG G 2 be the vertex-dsjot uo of the graphs G ad G 2 ThegraphG G 2 s obtaed from G G 2 by addg all possble edges from vertces of G to vertces of G 2 ;thats,g G 2 = G G 2 [5] The Laplaca matrx of the graph G s the matrx LG = DG AG,whereAG ad DG are the 0, -adjacecy matrx ad the dagoal matrx of the vertex degrees of G, respectvely The ormalzed Laplaca matrx of G s defed as L = DG /2 LGDG /2,whereDG /2 s the matrxwhch s obtaedby takg /2 power of each etry of DGTheLaplacaegevaluesadtheormalzed Laplaca egevalues of G are the egevalues of LG ad L, respectvelyletμ μ 2 μ be the Laplaca egevalues ad λ λ 2 λ the ormalzed Laplaca egevalues of GNotethatμ =0, λ =0,adthemultplctes of these zero egevalues are equal to the umber of coected compoets of G; see[6, 7] For more detals o Laplaca ad ormalzed Laplaca egevalues, see [6, 8 0] The umber of spag trees, tg, of the graph G s equaltothetotalumberofdstctspagsubgraphsofg that are trees Ths quatty s also kow as the complexty of G adgvebythefollowgformulatermsofthelaplaca egevalues [5]: t G = = μ It s well kow that the umber of spag trees of G s also expressed by the ormalzed Laplaca egevalues as [5, 6] t G = = d 2m = λ 2 Now, we gve some kow upper bouds o tg: Grmmett []: t G 2m, 3 2 Groe ad Merrs [2]: t G = d 2m, 4

2 2 The Scetfc World Joural 3 Nosal [3]: forr-regular graphs, t G r, 5 4 Cvetkovć et al see [5, page 222]: t G 2 m, 6 where m s the umber of edges of G, 5 Das [4]: 6 Zhag [5]: t G 2m Δ, 7 t G + a a 2m, 8 where a = 2m/2m 2 /2, 7 Feg et al [6]: t G Δ + t G M +2m Δ L et al [7]: 9 Bozkurt [8]: 2m Δ, 9 /2, 0 t G δ 2m Δ 3 δ, 3 t G + b b = d 2m, where b = Δ / 2Δ /2, 0 Das et al [9]: t G < t G 2m Δ δ 2m Δ δ, Δ +Δ m Δ Δ 2 3, t G /2 M + 2m [4m2 ] 3 I [] Grmmet pots out that 3 geeralzes 5 Groe ad Merrs [2] observed that, by the applcato of arthmetc-geometrc mea equalty, 4 leads to3 Das [4] stated that 7 ssharpfors or K,but3, 4, 5, ad 6 are sharp for oly K I[7] Letaldcatedthat s sharp for S, K, G K K K or K e,but3 s sharp for oly K ad 7 ad9 aresharpfors or K However, Das et al [9] provedthat sottruefork I [5, 6, 8] the authors showed that 8 sbettertha3, 9sbettertha7ad0, ad 2sbettertha4 For more bouds ad the relatos betwee the umber of spag trees ad the structural parameters of graphs such as coectvty, chromatc umber, depedece umber, ad clque umber, see [7, 9] We orgaze ths paper the followg way I Secto 2, we gve some prevously kow results whch wll be eeded later I Secto 3, we obta some bouds for the umber of spag trees of coected graphs terms of the umber of vertces, the umber of edgesm, maxmum vertex degree Δ, mmum vertex degree δ, frstzagrebdex M, ad Radć dex R Wealsoshowedthatsomeof our results o coected bpartte graphs mprove the bouds 9ad0forthesegraphs 2 Lemmas I ths secto, we gve some useful lemmas whch wll be used later Frstly, we troduce a auxlary quatty for a graph G as α= 2 [Δ +δ+ Δ δ 2 +4Δ ], 4 where Δ ad δ arethemaxmumadthemmumvertex degree of G,respectvely The result the followg lemma s also kow as Kober s equalty Lemma see [20] Let x,x 2,,x N be oegatve umbers ad let β= N N = x, γ = N = x /N be ther arthmetc ad geometrc meas, respectvely The 5 N N x j <j x 2 β γ N x x j 2 <j 6 Moreover, equalty 6 holdsfadolyfx =x 2 = = x N Lemma 2 see [2] Let G be a graph wth vertces ad ormalzed Laplaca matrx L wthout solated vertces The = = λ = tr L =, λ 2 = tr L 2 =+2 d d j V V j 7

3 The Scetfc World Joural 3 Lemma 3 see [8] Let G be a graph wth vertces ad wthout solated vertces The λ =λ 2 = =λ f ad oly f G s a complete graph K Lemma 4 see [8] Let G be a coected graph wth >2 vertces The λ 2 =λ 3 = = λ f ad oly f G K or G K p,q Note that, the Laplaca egevalues of a bpartte graph G cocde wth ts sgless Laplaca egevalues, that s, egevalues of the sgless Laplaca matrx DG+AG [9, 0, 22] Thus, oe ca arrve at the followg result Lemma 5 see [23, 24] Let G be a coected bpartte graph wth 3vertces ad let Δ be the maxmum vertex degree of GThe μ α Δ + 8 wth ether equaltes f ad oly f G s a star graph S Lemma 6 see [9] Let G be a graph wth vertces The μ, wth equalty f ad oly f G s dscoected Lemma 7 see [4] Let G be a coected graph wth 3 vertces The μ 2 =μ 3 = = μ f ad oly f G K or G S or G K Δ,Δ 3 Ma Results Recetly, Das et al [9] establshed upper ad lower bouds o tg applyg Kober s equalty to Laplaca egevalues of a coected graph GWeowcosderKober sequalty for the ormalzed Laplaca egevalues of G order to preset some bouds o tg Theorem 8 Let G be a coected graph wth vertces, m edges, ad RadćdexR The t G = d 2m [ 2 +2R ] t G = d 2m [ 2 + 2R ] /2 /2, 9 20 Moreover, equaltes 9 ad 20 holdfadolyfg K Proof Takg N=, x =λ 2,ad =,2,, Lemma,weget <j λ λ j 2 = λ2 2/ λ = 2 <j λ λ j 2 By the proof of Theorem 7 [9]adLemma 2,wehave <j λ λ j 2 = = λ 2 λ = = + 2 V V j d d j 2 = + 2R 2 The, combg 2wth ths ad2, we get + 2R 2 Ths mples that +2R / 2mt G = d + 2R 2 2/ 2mt G = d 2 +2R, 2/ 2mt G = d 2 +2R HeceweobtathefrstpartofthetheoremNowwe suppose that the equaltes 9 ad20 holdthe,by Lemma, wehaveλ =λ 2 = = λ Therefore, from Lemma 3,wegetthatG K Coversely, we ca easly see that the equaltes 9ad 20holdforthecompletegraphK We ow cosder the above theorem for coected bpartte graphs Theorem 9 Let G be a coected bpartte graph wth > 2 vertces, m edges, ad RadćdexR The t G = d m [ R 4] t G = d m [ 3 /2 +2R 4] Moreover, equaltes 25 hold f ad oly f G K p,q /2, 25

4 4 The Scetfc World Joural Proof Takg N=, x Lemma,wehave =λ 2,ad = 2,, 2 <j λ λ j 2 =2 λ2 2/ 3 λ = <j λ λ j 2 Sce G s bpartte, we also have λ = 2 [6] The, by Lemma 2,weget λ λ j 2 2 <j = =2 λ 2 λ =2 = + 2 V V j d d j 4 2 = + 2R 4 2 Therefore, combg 26 wththsad2, we arrve at + 2R R 4 2 mt G = d 2/ + 2R 4 2 Ths mples that mt G = d 2/ R 4, 2/ mt G = d 3 +2R Hecewegettheequaltes25 Now we suppose that the equaltes 25holdThe,byLemma,wehaveλ 2 =λ 3 = = λ Therefore, by Lemma 4, wecocludethatg K p,q Coversely, we ca easly see that the equaltes 25 hold for the complete bpartte graph K p,q We ow preset the mprovemet of the results obtaed [6]forbparttegraphs Theorem 0 Let G be a coected bpartte graph wth 3 vertces ad m edges ad let α be gve by 4The t G α α 2m 30 wth equalty f ad oly f G S Proof From ad Lemmas 5 7, oe ca prove30 a smlar way to the proof of Theorem [6] Remark From Lemma 5,wehaveμ α Δ +Theby the proof of Theorem [6], oe may coclude that 30 mproves 9forbpartte graphs Theorem 2 Let G be a coected bpartte graph wth 3 vertces, m edges, ad frst Zagreb dex M ad let α be gve by 4The t G M +2m α 2 /2 wth equalty f ad oly f G S 3 Proof From ad Lemmas 5 7, the proof of3 ca be easly gve a smlar way to the proof of Theorem 2 [6] Remark 3 From Lemma 5,wehaveμ α Δ +The by the proof of Theorem 2 [6], oe may coclude that 3mproves0forbparttegraphs Remark 4 Byusgthesmlarmaer[6], oe ca easly show that 30 sbettertha3 Moreover, f we ca obta a ew boud μ α α Δ +,theweca mprove the bouds 30ad3 Example 5 Let G be a graph wth vertex set VG = {V, V 2, V 3, V 4, V 5 } ad edge set E G ={e = V V 2,e 2 = V V 3, e 3 = V 2 V 4,e 4 = V 2 V 5,e 5 = V 4 V 5 } 32 For ths graph, tg s equal to 3 At rouded three decmal places, the bouds 8, 9,, 2, 3, ad 9gvetG 5659, tg 6400, tg 6250, tg 5224, tg 5859, tg 7200, tg 6422, adtg 504, respectvely Ths shows that the boud 9 s the best amog the metoed upper bouds for tg But geeral sese, they are ot comparable Coflct of Iterests The authors declare that there s o coflct of terests regardg the publcato of ths paper Ackowledgmets The authors are partally supported by TUBITAK ad the Offce of SelçukUverstyResearchProjectBAP Refereces [] I Gutma ad K Ch Das, The frst Zagreb dex 30 years after, MATCH Commucatos Mathematcal ad Computer Chemstry, o 50, pp 83 92, 2004

5 The Scetfc World Joural 5 [2] B Bollobás ad P Erdös, Graphs of extremal weghts, Ars Combatora,vol50,pp ,998 [3] M Cavers, S Fallat, ad S Krklad, O the ormalzed Laplaca eergy ad geeral Radć dexr of graphs, Lear Algebra ad Its Applcatos,vol433,o,pp72 90,200 [4] G Yu ad L Feg, Radć dex ad egevalues of graphs, Rocky Mouta Mathematcs,vol40,o2,pp73 72, 200 [5] D M Cvetkovć, M Doob, ad H Sachs, Spectra of Graphs,vol 87 of Pure ad Appled Mathematcs, Academc Press, New York, NY, USA, 980 [6]FRKChug,Spectral Graph Theory, CBMSLectureNotes, AMS, Provdece, RI, USA, 997 [7] M Fedler, Algebrac coectvty of graphs, Czechoslovak Mathematcal Joural, vol 23, o 98, pp , 973 [8] K Ch Das, A D Gügör, ad Ş B Bozkurt, O the ormalzed Laplaca egevalues of graphs, Ars Combatora I press [9] R Merrs, Laplaca matrces of graphs: a survey, Lear Algebra ad Its Applcatos, vol 97-98, pp 43 76, 994 [0] R Merrs, A survey of graph Laplacas, Lear ad Multlear Algebra,vol39,o-2,pp9 3,995 [] G R Grmmett, A upper boud for the umber of spag trees of a graph, Dscrete Mathematcs, vol6,o4,pp , 976 [2] R Groe ad R Merrs, A boud for the complexty of a smple graph, Dscrete Mathematcs, vol 69, o, pp 97 99, 988 [3] E Nosal, Egevalues of graphs [MS thess], Uversty of Calgary, 970 [4] K Ch Das, A sharp upper boud for the umber of spag trees of a graph, Graphs ad Combatorcs, vol23,o6,pp , 2007 [5] X-D Zhag, A ew boud for the complexty of a graph, Utltas Mathematca,vol67,pp20 203,2005 [6] LFeg,GYu,ZJag,adLRe, Sharpupperboudsfor the umber of spag trees of a graph, Applcable Aalyss ad Dscrete Mathematcs,vol2,o2,pp ,2008 [7] J L, W C Shu, ad A Chag, The umber of spag trees of a graph, Appled Mathematcs Letters,vol23, o3,pp , 200 [8] Ş B Bozkurt, Upper bouds for the umber of spag trees of graphs, Iequaltes ad Applcatos, vol 202, artcle 269, 7 pages, 202 [9]KChDas,ASCevk,adINCagul, Theumberof spag trees of a graph, Iequaltes ad Applcatos,vol203,artcle395,3pages,203 [20] H Kober, O the arthmetc ad geometrc meas ad o Hölder s equalty, Proceedgs of the Amerca Mathematcal Socety,vol9,pp ,958 [2] P Zumste, Comparso of spectral methods through the adjacecy matrx ad the Laplaca of a graph [Dploma thess],eth Zürch, 2005 [22] D Cvetkovć, P Rowlso, ads K Smć, Sgless Laplacas of fte graphs, Lear Algebra ad Its Applcatos,vol423,o, pp 55 7, 2007 [23] Y Che ad L Wag, Sharp bouds for the largest egevalue of the sgless Laplaca of a graph, Lear Algebra ad Its Applcatos,vol433,o5,pp908 93,200 [24] Y Zhag, X Lu, B Zhag, ad X Yog, The lollpop graph s determed by ts Q-spectrum, Dscrete Mathematcs, vol 309, o 0, pp , 2009

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