THE TRUNCATED RANDIĆ-TYPE INDICES
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1 Kragujeac J Sc 3 ( UDC 547:54 THE TUNCATED ANDIĆ-TYPE INDICES odjtaba horba, a ohaad Al Hossezadeh, b Ia uta c a Departet of atheatcs, Faculty of Scece, Shahd ajae Teacher Trag Uersty, Tehra, 785-3, I Ira b Departet of atheatcal Scece, Sharf Uersty of Techology, Tehra, , I Ira c Faculty of Scece, Uersty of Kragujeac, P O Box 0, Kragujeac, Serba (eceed Jauary 30, 009 ABSTACT For a graph = (V,E, the geeral adć dex s defed as ( ( = deg ( udeg (, where s a arbtrary real uber I ths paper we u E troduce the trucated erso of ths dex, ( U (, whch the suads pertag to a selected subset U of ertces of are abadoed The trucated hgher-order adć dex ( U (,, s also put forward The dces graphs are coputed ( U ( ad ( U ( of cha Itroducto I 975 la ANDIĆ [] proposed a topologcal dex / order to ease the extet of brachg of the carbo-ato seleto of satated hydrocarbos Already adć otced that there s a good correlato betwee hs dex ad seeral physco-checal propertes of alaes, such as bolg pot, chroatographc reteto te, ethalpy of forato, paraeters the Atoe equato for apor presse, etc Later, 998 BOLLOBÁS ad EDŐS [] geeralzed ths dex by replacg the expoet / by ay real uber For a graph = ( V(, E(, the geeral adć dex ( of s thus defed as the su of the ters ( deg ( udeg ( all edges u of, where deg ( u deotes the degree of the ertex u of, that s oer
2 48 u E ( ( ( = deg ( udeg ( Let U = { u, u,, u } be a subset of V( We ow defe a ew erso of the geeral adć dex ad ae t the trucated adć dex ( U as ( ( u, u,, u u ( = deg ( deg ( u E ( u, U e, ( U ( ( deg ( deg ( u = u E ( u, U The -th order coectty dex of a orgac olecule whose olecule graph s s defed by [3] ( = deg ( deg ( Ldeg ( where the suato rus oer all paths of legth Slarly we defe the trucated -th order coectty dex as: ( u, u,, u ( = U,,, deg ( deg ( L deg ( e, ( U ( = L U,,, deg ( deg ( deg ( If U deotes the subgraph obtaed by deletg the ertces u, u,, u fro the graph, the t s edately see that ( ( U u u E ( U ( = deg ( deg ( ad ( U ( = ( U deg ( deg ( Ldeg (
3 49 a esults ad Dscusso I ths secto we copute the trucated adć dex of the cha graphs The we use ths ethod to copute the geeral adć dex for a fte class of aostar dedrers Let ( be soe graphs ad V( A cha graph deoted by = (,,,,, s obtaed fro the uo of the graphs, =,,,, by addg these edges ( -, see Fg The ( = V V ( ad ( = ( E E ( = = Fg The cha graph = (,,,,, It s worth otg that the aboe specfed class of cha graphs ebraces, as specal cases, all trees (aog whch are the olecular graphs of alaes ad all ucyclc graphs (aog whch are the olecular graphs of oocycloalaes Also the olecular graphs of ay polyers ad dedrers are cha graphs Lea Suppose that = (,,,,,,, s a cha graph The: ( (,,,,,,, s coected f ad oly f ( are coected ( deg (a a V( ad a deg ( a = deg ( a a=,, = deg ( a a=, Proof s straghtforward Theore The trucated adć dex of the cha graph = (,,, (, u,, u s:
4 50 ( u,, u ( u,, u, ( u,, u, = ( ( ( = u E( u u,, u (( deg ( deg ( u (( ( deg ( deg ( Proof By usg the defto of the trucated adć dex oe ca see that ( u,, u ( ( deg ( deg ( = u u E ( u, u,, u = ( deg ( deg ( u ( deg ( udeg ( u E ( u E ( u, u,, u, u, u,, u, ( ( deg ( (( deg ( u deg ( deg ( u u E( u E( u u,, u u u,, u ( ( deg ( ( deg ( ( u,, u, ( u,, u, = ( = u E( u u,, u ( ( ( deg ( deg ( u ( ( deg ( ( deg ( Theore 3 If 3 ad,, u,, u, the for = (,,,,,,, t holds: ( ( deg ( ( deg ( (( deg ( ( deg ( L ( ( ( ( u,, u ( u,, u, = u ( ( deg ( deg ( = = u E( u u,, u = ( deg ( deg ( Proof s aalogous as that of Theore 3 Applcatos
5 5 Exaple Cosder the graph show Fg It s easy to see that ad so, ( 3 3 ( =, ( 3 ( = ( = ( = ( = 5 3 ( ( ( for, j 3, j (, (, j 3 ( = ( = 3 Fg The graph of aostar for = Cosder ow the cha graph = (, H,, u, show Fg (for = ad Fg 3, respectely It s easy to see that H ( - ad = (, H,, u = (, H,, u = (, H,, u = (, H,, u The by usg Theore, we hae the followg relatos: ( = ( ( H 3 4 ( ( u ( ( (, u ( = ( ( H 3 4 ( = ( ( H 3 4 ( ( (, u ( = ( ( H 3 4 ( ( (, u
6 5 Suato of these relatos yelds ( ( u (, u = ( = ( ( H ( H ( (3 4 Fg 3 The cha graph ad the labelg of ts ertces ad so by usg Exaple, t s easy to obta ( (, ( = ( ( ( ( (3 4 ( = ( (4 3 (0 4 I other words we arred at the followg: Theore 4 Cosder the cha graph = (, H,, u, show Fg 3 The, ( ( = ( (4 3 (0 4
7 53 Fg 4: The graph of the aostar dedrer D Corollary 4 Cosder the aostar dedrer D, show Fg 4 The, D = ( ( ( (4 3 (0 4 where s the uber of repetto of the fraget Theore 5 For the cha graph = (,,,,,,,, u,,,,, j, p, q, h ( u,, ( u,,, ( ( = = h= = 0 deg ( deg ( Ldeg ( whch the lasr suato we hae the codtos,,, u,, u r There s a arable j such that =,, = ad for all p j ad q j h we hae V( ad V( j j h h p q, h respectely Proof By the codtos of the theore we hae: ( u,, ( =,,, u,, deg ( deg ( deg ( = =,,, u,,, j V( deg ( deg ( Ldeg (
8 54 =,,, u,, j V( j : j= deg ( deg ( Ldeg ( =,,, u,, j : j=, j = p j: p V( q j : q V( deg ( deg ( Ldeg ( =,,, u,, j : j=, j =, j = p j: p V( q j : q V( deg ( deg ( Ldeg ( =,,, u,, j : j=, j =,, j = p j: p V( q j : q V( deg ( deg ( Ldeg ( deg ( deg ( deg ( h ( u,,, = ( = h= 0 = L,,, u,, j : j=,, j h= h p j: p V( q j h: q V( h Exaple Cosder the graph depcted Fg We hae the results dsplayed Table, = t whch (,,, ad c = j V( t, j j : j = deg ( deg ( Ldeg ( deg ( deg ( Ldeg ( j V(, j j : j = deg ( deg ( Ldeg ( j : j=, j = ( ( (, ( c
9 Table Now cosder the cha graph = (, H,, u show Fg 3 Oe ca see that for 7 ( = ( ( H c ( ( u ( = ( ( H c ( ( (, u ' ( = ( ( H c ( ( (, u ' ( = ( ( H c ( ( (, u ' where for - ad = (, H,, u, c = deg ( deg ( Ldeg ( j V(, j j : j= deg ( deg ( Ldeg ( j V( H, j j j : j= u deg ( deg ( Ldeg ( j, j j : j= u, j = Because 7, t s easy to see that c = c for u u Therefore, by suato of these relatos we get:
10 5 ( = ( ( ( c ( c ( (, By ths we hae proe the followg: Theore For the aostar dedrer show Fg 4, ( = ( ( ( ( c ( (, whch for = ( t,, u, u, c s sae as aboe, ad u u The -coectty dces of for 7 are ge Table ( ( Table EFEENCES ANDIĆ, O characterzato of olecular brachg, J A Che Soc 97 ( B BOLLOBÁA, P EDŐS, raphs of extreal weghts, Ars Cob 50 ( L B KIE, L H HALL, olecular Coectty Chestry ad Drug esearch, Acadec Press, New Yor, 97
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