Malaya Joural of Matematik Vol 6 No 349-353 08 https://doiorg/06637/mjm060/0008 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 36300 Novi Pazar Serbia *Correspodig author: igor@elfakiacrs Article History: Received 4 November 07; Accepted 9 Jauary 08 State Cotets Itroductio 349 Prelimiaries 350 3 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
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
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)
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 36 37 ad 38 with those give i Theorems 30 3 ad 3 35
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