Relationship between augmented eccentric connectivity index and some other graph invariants

Transcription

1 Iteratioal Joural of Advaced Mathematical Scieces, () (03) 6-3 Sciece Publishig Corporatio wwwsciecepubcocom/idexphp/ijams Relatioship betwee augmeted eccetric coectivity idex ad some other graph ivariats Nilaja De Departmet of Basic Scieces ad Humaities (Mathematics),Calcutta Istitute of Egieerig ad Maagemet, Kolkata, West Begal, Idia Abstract The augmeted eccetric coectivity idex of a graph which is a geeralizatio of eccetric coectivity idex is defied as the summatio of the quotiets of the product of adjacet vertex degrees ad eccetricity of the cocered vertex of a graph I this paper we established some relatioships betwee augmeted eccetric coectivity idex ad several other graph ivariats like umber of vertices, umber of edges, maximum vertex degree, miimum vertex degree, the total eccetricity idex the first Zagreb idices ad the secod multiplicative Zagreb idex Keywords:Vertex Degree, Eccetricity, Zagreb Eccetricity Idices Itroductio I this paper we are cocered oly with fiite simple coected graph ie fiite coected graph without self loops ad multiple edges For a graph G let the set of all vertices ad edges are deoted by V(G) ad E(G) respectively Also let ad m be, respectively, the umber of vertices ad edges of G For all v belogs to V(G) let be the umber of first eighbor ie degree of the vertex v Also let max : v V( G) ad mi : v V( G) be the maximum ad the miimum vertex degree, respectively Let the distace betwee ay two vertices of V(G), is equal to the legth of the shortest path coectig them ad deoted by d(u,v) Also for a give vertex of V(G) its eccetricity is the largest distace from that vertex to ay other vertices of G ie ( v) max{ d( v, x) : x V( G)} The radius ad diameter of the graph are the smallest ad largest eccetricity amog all the vertices of G respectively [] ie r r( G) mi{ ( v) : vv( G)} ad d d( G) max{ ( v) : v V( G)} Also the total eccetricity of a graph, deoted by ( G), is the sum of eccetricities of all the vertices of G [] The oldest ad most popular graph ivariat, the classical Zagreb idices, itroduced by Gutma ad Triajstić are defied as [5] M ( G) deg( ) v, vv ( G) M ( G) deg( u) ( u, v) E ( G) where is the degree of the vertex v The Zagreb idices were subject to a large umber of mathematical studies [4] Todeschii et al [6] have itroduced the multiplicative variats of additive graph ivariats, which applied to the Zagreb idices would lead to the first ad secod Multiplicative Zagreb Idex which are defied as ( G) deg( ) v, vv ( G) ( G) deg( u) ( u, v) E( G)

2 Iteratioal Joural of Advaced Mathematical Scieces 7 The properties of these multiplicative Zagreb idices for trees were studied by Gutma [0] ad some properties, bouds were studied by Liu ad Zhag [0] I recet years a umber of graph ivariats related to eccetricity have bee derived ad studied Oe of them the eccetric coectivity idex of a graph G was proposed by Sharma, Goswami ad Mada [] ad is defied as ac ( G) ( v) vv ( G) A more geeralizatio of the coective eccetric idex kow as augmeted eccetric coectivity idex was itroduced by Gupta, Sigh ad Mada [8] ad is defied as ac ( G) vv ( G) Mv () () v where Mv () deotes the product of degrees of all eighbors of vertex v i ie M ( v) deg( u) This idex is ( u, v) E( G) relatively ew though subject to some chemical as well as mathematical studies From above defiitio of augmeted eccetric coectivity idex it is evidet that, as the degrees are take over the eighborhoods ad the multiplied ad agai the reciprocal of eccetricity is cosidered for a vertex, so the cotributio of a vertex to this idex is both olocal ad o-lieardifferet properties of augmeted eccetric coectivity idex have bee studied by Došlić ad Saheli (0)[6], Sedlar (0)[], De (0)[3] The problem of fidig extremal properties of some topological idices so that to establish some iequalities ivolvig differet graph ivariats have bee studied by differet researchers of which oly some recet results are metioed here [, 4, 5, 0, 7] The aim of this paper is to ivestigate similar extremal properties for multiplicative augmeted eccetric coectivity idex I this paper we first establish some upper ad lower bouds of Mv () ad the usig those some sharp lower ad upper bouds of augmeted eccetric coectivity idex is give i terms of differet graph ivariats icludig the umber of vertices (), umber of edges (m), radius (r), diameter (d), maximum vertex degree ( ), miimum vertex degree (δ), the total eccetricity idex ( ( G) ) the first Zagreb idices (M (G) ) ad the secod multiplicative Zagreb idex ( ( G) ) vv ( G) Mai results The First we fid some upper ad lower bouds of summatio of degrees of all the eighbors of the vertices of G is Mv () vv ( G) Propositio : Let G be a simple coected graph with vertices ad ( G) deotes the first Zagreb idex of G, the M ( v) ( G) with equality if ad oly if all the vertices of G are of same degree Proof Usig iequality betwee arithmetic ad geometric mea, we have v V ( G) v V ( G) M ( v) M ( v) deg( ) v vv ( G) ( u, v) E( G) deg( v ( ) deg( ) ) ( ) M v v G ( ) vv ( G) ie v V G with equality if ad oly if all the vertices of G are of same degree Here ( G) is the secod multiplicative Zagreb idex of G ad is defied as [9]

3 8 Iteratioal Joural of Advaced Mathematical Scieces ( u, v) E( G) vv ( G) ( G) deg( u) Propositio : Let G be a simple coected graph with vertices ad M ( G) deotes the first Zagreb idex of G, the M ( v) M ( G) with equality if ad oly if G K vv ( G) Proof Let () v deotes the sum of degrees of all the eighbors of the vertex v, so that So obviously from the defiitio of Mv () ( v) deg( u) ( u, v) E( G) Thus, M ( v) ( v) M( G) vv ( G) vv ( G) M( v) ( v) for all v V ( G) with equality if ad oly if for all v V ( G) Here M ( ) G is the first Zagreb eccetricity idex of G ad is defied as [4] M ( G) ( v) vv ( G) vv ( G) Propositio 3: Let G be a simple coected graph with vertices, the vv ( G) M () v with equality if ad oly if all the vertices of G are of same degree Proof Usig the iequality betwee arithmetic ad geometric mea, we have deg( ) deg( u) deg( u) v ( u, v) E( G) ( u, v) E( G) ie () v Mv () () Now sice,, so Thus from (), we ca write u ( u, v) E( G) ( v) deg( ) Mv () from where the desired result follows with equality if ad oly if all the vertices of G are of same degree Propositio 4: Let G be a simple coected graph with vertices, the m M ( v) ( ) vv ( G) with equality if ad oly if all the vertices of G are of same degree

4 Iteratioal Joural of Advaced Mathematical Scieces 9 Proof We have [4] for all v V ( G), so that ( v) m ( ) ( v) m ( ) m Thus, ( v) m So from () we have, vv ( G) vv ( G) ( v) m M ( v) ( ) which is our desired result I the above iequality equality holds if ad oly if all the vertices of G are of same degree Now we fid some upper ad lower bouds of augmeted eccetric coectivity idex Theorem : Let G be a simple coected graph with vertices ad m edges, the ac ( G) d r with equality if ad oly if G all the vertices of G are of same degree ad eccetricity Proof Sice, for all v V ( G) ad Mvis () the product of all the eighbors of the vertex v, we have Mv (), for all v V ( G), with equality if ad oly if G is a regular graph Similarly sice r () v d, for all v V ( G), from the defiitio of augmeted eccetric coectivity idex, we have d ac ( G) r which is our desired result Obviously i the above iequality equality holds if ad oly if all the vertices are of same degree ad eccetricity Theorem : Let G be a simple coected graph with vertices ad m edges, the ac G ( G) ( ) ( G) with equality if ad oly if all the vertices of G are of same degree ad eccetricity Proof To prove this theorem we use Chebyschev s iequality as follows ac Mv ( ) ( G) M ( v) ( v) ( v) vv ( G) vv ( G) vv ( G) with equality if ad oly if all the vertices of G are same degree ad eccetricity Now usig the iequality betwee arithmetic ad harmoic mea, we have

5 30 Iteratioal Joural of Advaced Mathematical Scieces vv ( G) ie () v vv ( G) vv ( G) () v ( v) ( G) with equality if ad oly if all the vertices of G are of same eccetricity So from above we ca write ac ( G) M ( v) () ( G) vv ( G) Now usig Propositio from above the desired result follows, with equality if ad oly if all the vertices of G are of same degree ad eccetricity Corollary : Let G be a simple coected graph with vertices ad m edges, the ( ) ac M G ( G) ( G) where M ( ) G is the first Zagreb idex of G I the above iequality, equality holds if ad oly if G is a path of legth oe Proof Usig propositio, the above result follows from relatio () Obviously equality holds i the above iequality if ad oly if G K Theorem 3: Let G be a simple coected graph, the with equality if ad oly if G K Proof Sice we have [3] () v for all v V ( G), with equality if ad oly if G K je for j,,, or G P4, so from the defiitio of augmeted eccetric coectivity idex, we ca write ac ( G) M ( v) (3) v V ( G) ac ( G) Now usig Propositio 3, the desired result follows from (3) The equality holds i the Propositio 3 if all the vertices are of same degree Thus i this theorem equality holds if ad oly if G K Corollary : Let G be a simple coected graph with vertices ad m edges, the with equality if ad oly if G K ac m ( G) ( ) ( ) Proof Usig Propositio 4, the above result follows from relatio (3) The equality holds i the Propositio 4 if ad oly if all the vertices are of same degree Thus i this result equality holds if ad oly if G K

6 Iteratioal Joural of Advaced Mathematical Scieces 3 Theorem 4: Let G be a simple coected graph, the ac ( ) ( G) with equality if ad oly if G K Proof Sice we have [3] ( v) D( v) for all v V ( G), with equality if ad oly if G K so ( ) ( v) D( v) vv ( G) vv ( G) Now sice, D( v) for all v V ( G), with equality if ad oly if G K, we have from above ( ) ( v) vv ( G) vv ( G) Agai sice, D( v) for all v V ( G), we ca write ( ) (4) () v vv ( G) So, usig defiitio of augmeted eccetric coectivity idex ad Propositio 3, the desired result follows from (4) Obviously i the above iequality equality holds if ad oly if G K Corollary 3: Let G be a simple coected graph with vertices ad m edges, the with equality if ad oly if G K Proof From Propositio 4, we have ac ( ) m ( G) ( ) m M ( v) ( ) with equality if ad oly if G is regular So from the defiitio of augmeted eccetric coectivity idex we ca write ac m ( G) ( ) () v vv ( G) Now usig (4), from above we get the desired result Sice the equality holds i the Propositio 4 if ad oly if all the vertices are of same degree Thus i this result equality holds if ad oly if G K 3 Coclusio I this paper, we have established sharp upper ad lower bouds foraugmeted eccetric coectivity idex i terms of graphparameters icludig the order, size, radius, diameter, maximum vertex degree, miimum vertex degree, the total

7 3 Iteratioal Joural of Advaced Mathematical Scieces eccetricity idex, the first Zagreb idicesad the secod multiplicative Zagreb idex, so that differet iequalities related to augmeted eccetric coectivity idex to some other graph ivariats with the respective equality coditio were establishedit may be iterestig to give the bouds for augmeted idex i terms of other graph ivariats Refereces [] PDakelma, WGoddard, Swart C S (004) The average eccetricity of a graph ad its subgraphs, Util Math, 65, pp4-5 [] N De (0) Some bouds of reformulated Zagreb idices, Applied Mathematical Scieces, 6,pp [3] N De (0) Augmeted Eccetric Coectivity Idex of Some Thor Graphs Iteratioal Joural of Applied Mathematical Research, (4),pp [4] N De (0) Bouds for the coective eccetric idex, Iteratioal Joural of Cotemporary Mathematical Scieces, 7(44): pp6-66 [5] N De (03), New Bouds for Zagreb Eccetricity Idices, Ope Joural of Discrete Mathematics,3(), pp [6] TDošlić, M Saheli (0) Augmeted Eccetric Coectivity Idex, Miskolc Mathematical Notes, (),pp49-57 [7] H Dureja, A KMada (007) Superaugmeted eccetric coectivity idices: ew geeratio highly discrimiatig topological descriptors for QSAR/QSPR modelig, Med Chem Res, 6,pp33 34 [8] SGupta, M Sigh, A K Mada (000) Coective eccetricity Idex: A ovel topological descriptor for predictig biological activity, Joural of Molecular Graphics ad Modellig, 8,pp8-5 [9] I Gutma (0) Multiplicative Zagreb Idices of Trees, Bulleti of Society of Mathematics Baja Luka,8,pp7-3 [0] J Liu, V Zhag(0) Sharp Upper Bouds for Multiplicative Zagreb Idices, MATCH Commuicatios i Mathematical ad i Computer Chemistry, 68: pp3-40 [] J Sedlar (0) O augmeted eccetric coectivity idex of graphs ad trees, MATCH Commuicatios i Mathematical ad i Computer Chemistry, 68, pp35-34 [] V Sharma, R Goswami, A K Mada (997) Eccetric coectivity idex: A ovel highly discrimiatig topological descriptor for structure-property ad structure-activity studies, Chemical Iformatio Modelig, 37, pp73-8 [3] B Zhou, Z Du, O eccetric coectivity idex, MATCH Commu Math Comput Chem, 63 (00),pp8 98 [4] K C Das, I Gutma, Some properties of the secod Zagreb idex, MATCH Commu Math Comput Chem, 5 (004),pp03 [5] I Gutma, N Triajstić, Graph theory ad molecular orbitals, Total π-electro eergy of alterat hydrocarbos, Chem Phys Lett, 7 (97), [6] R Todeschii, V Cosoi, New local vertex ivariats ad molecular descriptors based o fuctios of the vertex degrees, MATCHCommu Math Comput Chem 64 (00) [7] N De, O multiplicative Zagreb eccetricity idices, South Asia J Math, 0, (6), pp