Some inequalities for the Kirchhoff index of graphs

Transcription

1 Malaya Joural of Matematik Vol 6 No Some iequalities for the Kirchhoff idex of graphs Igor Milovaovic * Emia Milovaovic Marja Matejic ad Edi Glogic Abstract Let G be a simple coected graph of order sequece of vertex degrees d d d > 0 ad Laplacia eigevalues µ µ µ > µ 0 With Π Π (G) i di we deote the multiplicative first Zagreb idex of graph ad K f (G) i the Kirchhoff idex of G I this paper we determie several lower ad upper bouds for K f depedig o some of the graph parameters such as umber of vertices maximum degree miimum degree ad umber of spaig trees or multiplicative Zagreb idex Keywords Kirchhoff idex Laplacia eigevalues (of graph) vertex degree AMS Subject Classificatio 05C 05C50 Faculty of Electroic Egieerig 8000 Nis Serbia Uiversity of Novi Pazar Novi Pazar Serbia *Correspodig author: igor@elfakiacrs Article History: Received 4 November 07; Accepted 9 Jauary 08 State Cotets Itroductio 349 Prelimiaries Mai results 350 Refereces 353 Itroductio Let G (V E) be a simple coected graph with vertices ad m edges with vertex degree sequece d d d > 0 ad d d d d If vertices i ad j are adjacet we write i j Further let A be the adjacecy matrix of G ad D diag(d d d ) the diagoal matrix of its vertex degrees The L D A is the Laplacia matrix of G Eigevalues of L µ µ µ > µ 0 form the so-called Laplacia spectrum of G Followig idetities are valid for (see []) m i ad di di M m i i M M (G) i di i where is the first Zagreb idex itroduced i [] More about this degree based topological idex oe ca be foud i [3 8] It is well kow that a coected graph G of order has t t(g) i c 08 MJM spaig trees I [9] (see also [0]) a multiplicative variat of the first Zagreb idex amed the first multiplicative Zagreb idex Π was itroduced It is defie as Π Π (G) di i I [] Klei ad Radic itroduced the otio of resistace distace ri j It is defied as the resistace betwee the odes i ad j i a electrical etwork correspodig to the graph G i which all edges are replaced by uit resistors The sum of resistace distaces of all pairs of vertices of a graph G is amed as the Kirchhoff idex ie K f (G) ri j i< j There are several equivalet ways to defie the resistace distace Gutma ad Mohar [] (see also [3]) proved that the Kirchhoff idex ca be obtaied from the o-zero eigevalues of the Laplacia matrix: K f (G) i Amog various idices i mathematical chemistry those based o the effective resistace ri j such as the Kirchhoff idex ad its geeralizatios have received a lot of attetio

2 Some iequalities for the Kirchhoff idex of graphs 350/353 i the literature as it tured out that they play a importat role i solvig problems i differet scietific disciplies such as molecular chemistry spectral graph theory etwork theory etc (see for example [4 ]) Cosiderig the fact that obtaiig the exact ad easy to compute formula for the Kirchhoff idex is ot always possible it is useful to kow approximatig expressios ie upper ad lower bouds ad correspodig extremal graphs I this paper we report several lower ad upper bouds for K f (G) of a coected (molecular) graph i terms of some structural graph parameters such as the umber of vertices (atoms) maximum vertex degree (valecy) miimal vertex degree ad graph ivariats such as umber of spaig trees t ad multiplicative first Zagreb idex Π Prelimiaries I this sectio we recall some iequalities for the Kirchhoff idex ad some aalytic iequalities for real umber sequeces that are of iterest for the subsequet cosideratios Let G be a simple coected graph with vertices I [] the followig iequality was proved with equality if ad oly if a a3 a a a Let a (ai ) i be positive real umber sequece I [4] (see also [5]) the followig was proved ai i d i i i i (6) i Theorem 3 Let G be a simple coected graph with 3 vertices The for ay real k with the property µ k > 0 holds k K f (G) ( )(t) ( ) α( ) k (3) with equality if ad oly if k ad G K Proof For : ai bi i R R / µ r r / µ the iequality (3) trasforms ito! ( ) i i ( ) α( ) µ µ i where α() ai! I the followig theorem we establish upper boud for K f (G) i terms of umber of spaig trees umber of vertices ad parameter k where k is a arbitrary real umber such that µ k > 0 ai bi ai bi (R r )(R r )α() (3) i! ai 3 Mai results Let a (ai ) ad b (bi ) i be two positive real umber sequeces with the properties 0 < r ai R < ad 0 < r bi R < I [] the followig iequality was proved i with equalities if ad oly if a a a Before we proceed let us defie oe special class of dregular graphs Γd (see [0]) Let N(i) be a set of all eighbors of the vertex i ie N(i) {k k V k i} ad d(i j) the distace betwee vertices i ad j Deote by Γd a set of all d-regular graphs d with diameter ad N(i) N( j) d for i j () K or G K or G with equality if ad oly if G K or G Γd The followig lower boud for K f (G) that depeds o umber of vertices the maximum degree ad the umber of spaig trees t was determied i [4]: () ( ) t K or G K with equality if ad oly if G i ( ) ai i ( )! ai ai jk ( ) jk 4 Let a a a > 0 be real umber sequece I [3] it was proved a a a a a a ( a a ) (4) with equality if ad oly if a a3 a a a ad q q a a a a a a a (5) a a a 350 Sice 0 < µ ad µ k > 0 it follows! ( ) i i (3) ( ) α( ) k For : ai i left had side of iequality (6) becomes!! ( ) ( ) i i i i

3 Some iequalities for the Kirchhoff idex of graphs 35/353 ad µ (see [7]) ad the iequality (35) trasforms ito µ µ ( ) t i ie! ( ) i i ( )(t) (33) From (3) ad (33) we obtai ( ) i Now cosider the fuctio g(x) x ( ) tx It was proved that it is mootoe icreasig for x ad x (t) (see [4]) Sice µ (see [8]) ad µ (t) we have that ( ) ( )(t) i k ( ) α( ) k ie ( )(t) ( ) α( ) i k i k wherefrom we get (3) Equality i (33) holds if ad oly if µ µ therefore equality i (3) holds if ad oly if k ad K I the ext theorem we establish lower boud for K f (G) depedig o structural graph parameters ad the umber of spaig trees t wherefrom we obtai (34) Equality i (35) holds if ad oly if µ µ hece equality i (34) holds if ad oly if G K or G K or G K (see [9]) Similarly the followig result ca be proved Theorem 33 Let G be a simple coected graph with 3 vertices The ( ) t (36) Theorem 3 Let G be a simple coected graph with 3 vertices The ( ) t (34) Equality holds if ad oly if G K or G K or G K with equality if ad oly if G K or G K or G K Proof Accordig to (4) we have that a a3 a ( )(a a3 a ) For ai µ i forms ito! is stroger tha () i the iequality (36) Theorem 35 Let G be a simple coected graph with vertices The µ µ ( ) ( ) (Π ) ( ) (37) Equality holds if ad oly if G K or G K or G Γd µ µ ( ) i I the followig theorem we determie lower boud for K f (G) i terms of umber of vertices maximum degree ( a a ) miimum degree ad topological idex Π ie ( ) i Remark 34 Sice i the above iequality tras- ( ) t i! µ µ Proof For ai di i a a the iequality (5) trasforms ito q (35) Obviously equality i (35) ie (34) is attaied if G K Therefore suppose that G 6 K I that case µ (see [6]) 35 i di i di q (38)

4 Some iequalities for the Kirchhoff idex of graphs 35/353 ie ie ( ) (Π ) di (4 ) i di (Π ) (39) i (3) From (39) ad () we obtai (37) Equality i (38) holds if ad oly if d d d d d d ie if ad oly if d d d d Equality i () is attaied if ad oly if G K or G K or K or G Γd hece equality i (37) holds if ad oly if G K or G K or G Γd Fially from () ad (3) we arrive at (30) Equality i (3) holds if ad oly if d d ie if ad oly if d d Equality i () is attaied if ad oly if G K or G K or G K or G Γd therefore equality i (37) holds if ad oly if G K or G K or G Γd By a similar procedure as i case of Theorem 35 the followig results ca be proved I a similar way as i case of Theorem 39 the followig statemets ca be proved Theorem 36 Let G be a simple coected graph with 3 vertices The Theorem 30 Let G be a simple coected graph with 3 vertices The ( ) Π ( ) ( ) Equality holds if ad oly if G K or G K or G K or G Γd ( ) ( ) (4 ) Π ( ) Equality holds if ad oly if G K or G K or G K or G Γd Theorem 37 Let G be a simple coected graph with 3 vertices The ( ) ( ) (4 ) Π ( ) Equality holds if ad oly if G K or G K or G Γd Theorem 38 Let G be a simple coected graph with 4 vertices The ( )( ) ( )( ) ( ) (4 ) Π Theorem 3 Let G be a simple coected graph with 4 vertices The ( )( ) Π ( ) ( )( ) ( ) Equality holds if ad oly if G K or G K or G K or G Γd ( ) Equality holds if ad oly if G K or G K or G K or G Γd Theorem 39 Let G be a simple coected graph with vertices The ( ) (Π ) ( ) (30) Equality holds if ad oly if G K or G K or G Γd Proof For ai di i a a the iequality (4) trasforms ito di i di i! (3) Theorem 3 Let G be a simple coected graph with 3 vertices The ( ) Π ( ) ( ) Equality holds if ad oly if G K or G K or G Γd Remark 33 Lower bouds for K f (G) give by (37) ad (30) deped o the same parameters ad topological idex Π Equalities are achieved uder the same coditios ie if ad oly if G K or G K or G Γd However these bouds are ot comparable Thus for example for K the iequality (37) is stroger tha (30) but for P the iequality (30) is stroger tha (37) for 5 The same applies whe compare iequalities from Theorems ad 38 with those give i Theorems 30 3 ad 3 35

5 Some iequalities for the Kirchhoff idex of graphs 353/353 Ackowledgmet This paper was supported by the Serbia Miistry of Educatio ad Techological Developmet Grats TR30 ad 3009 [6] [7] Refereces [] [] [3] [4] [5] [6] [7] [8] [9] [0] [] [] [3] [4] [5] F R K Chug Spectral Graph Theory Amer Math Soc Providece 997 I Gutma ad N Triajstic Graph theory ad molecular orbitals Total π-electro eergy of alterat hydrocarbos Chem Phys Lett 7 (97) B Borovic ai K C Das B Furtula ad I Gutma Zagreb idices: Bouds ad Extremal graphs I: Bouds i Chemical Graph Theory Basics (I Gutma B Furtula K C Das E Milovaovic I Milovaovic Eds) Mathematical Chemistry Moographs MCM 9 Uiv Kragujevac Kragujevac 07 pp B Borovic ai K C Das B Furtula ad I Gutma Bouds for Zagreb idices MATCH Commu Math Comput Chem 78 (07) 7 00 I Gutma ad K C Das The first Zagreb idex 30 years after MATCH Commu Math Comput Chem 50 (004) 83 9 S Nikolic G Kovac evic A Milic evic ad N Triajstic The Zagreb idices 30 years after Croat Chem Acta 76 (003) 3 4 DW Leea S Sedghib ad N Shobec Zagreb Idices of a Graph ad its Commo Neighborhood Graph Malaya J Mat 4(3) (06) A Ghalavad ad A R Ashrafi Extremal trees with respect to the first ad secod reformulated Zagreb idex Malaya J Mat 5 (3)(07) R Todeschii D Ballabio ad V Cosoi Novel molecular descriptors based o fuctios of ew vertex degrees I: I Gutma ad B Furtula (Eds) Novel Molecular Structure Descriptors Theory ad Applicatios I (pp 73 00) Mathematical Chemistry Moographs MCM 8 Uiv Kragujevac Kragujevac 00 R Todeschii ad V Cosoi New local vertex ivariats ad molecular descriptors based o fuctios of the vertex degrees MATCH Commu Math Comput Chem 64() (00) D J Klei ad M Radic Resistace distace J Math Chem (993) 8 95 I Gutma ad B Mohar The quasi Wieer ad the Kirchhoff idices coicide J Chem If Comput Sci 36 (996) H Y Zhu D J Klei ad I Lukovits Extesios of the Wieer umber J Chem If Comput Sci 36 (996) K C Das O the Kirchhoff idex of graphs Z Naturforsch 68a (03) I Gutma B Furtula K C Das E Milovaovic ad I Milovaovic (Eds) Bouds i Chemical Graph Theory 353 [8] [9] [0] [] [] [3] [4] [5] [6] [7] [8] [9] Basics Mathematical Chemistry Moographs MCM 9 Uiv Kragujevac Kragujevac 07 J Liu J Cao X F Pa ad A Elaiw The Kirchhoff idex of hypercubes ad related complex etworks Discr Dyam Natur Sci (03) Article ID I Milovaovic I Gutma ad E Milovaovic O Kirchhoff ad degree Kirchhoff idices Filomat 9(8) (05) I Z Milovaovic ad E I Milovaovic O some lower bouds of the Kirchhoff idex MATCH Commu Math Comput Chem 78 (07) I Z Milovaovic ad E I Milovaovic Bouds of Kirchhoff ad degree Kirchhoff idices I: Bouds i Chemical Graph Theory Maistreams (I Gutma B Furtula K C Das E Milovaovic I Milovaovic Eds) Mathematical Chemistry Moographs MCM 0 Uiv Kragujevac Kragujevac 07 pp 93 9 J L Palacios Some additioal bouds for the Kirchhoff idex MATCH Commu Math Comput Chem 75 (06) B Zhou ad N Triajstic A ote o Kirchhoff idex Chem Phys Lett 455 (008) 0 3 M Bieracki H Pidek ad C Ryll Nardzewski Sur ue iegalite etre des itegrales defiies A Uiv Mariae Curie Sklodowska A 4 (950) 4 V Cirtoaje The best lower boud depeded o two fixed variables for Jese s iequality with ordered variables J Ieq Appl 00 (00) Article ID 858 B Zhou I Gutma ad T Aleksic A ote o the Laplacia eergy of graphs MATCH Commu Math Comput Chem 60 (008) H Kober O the arithmetic ad geometric meas ad o Ho lder s iequality Proc Amer Math Soc 9 (958) J S Li ad Y L Pa A ote o the secod largest eigevalue of the Laplacia matrix of a graph Li Multili Algebra 48 (000) 7 M Fiedler Algebraic coectivity of graphs Czech Math J 37 (987) R Merris Laplacia matrices of graphs: A survay Li Algebra Appl (994) K C Das Sharp upper boud for the umber of spaig trees of a graph Graphs Comb 3 (007) 65 63????????? ISSN(P): Malaya Joural of Matematik ISSN(O):3 5666?????????