ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONS

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1 International Jornal of Advanced Research in Engineering Technology (IJARET Volme 7, Isse 3, May Jne 06, pp 5 4, Article ID: IJARET_07_03_003 Available online at Jornal Impact Factor (06: 8897 (Calclated by ISI wwwjifactorcom ISSN Print: ISSN Online: IAEME Pblication ZAREB INDICES AND ZAREB COINDICES OF SOME RAPH OPERATIONS Assistant Professor, Mathematics, Sri Shanmgha College of Engineering Technology, Salem [Dt] TamilNad, India ABSTRACT A topological index is the graph invariant nmerical descriptor calclated from a moleclar graph representing a molecle Classical Zagreb indices the recently introdced Zagreb coindices are topological indices related to the atom atom connectivity of the moleclar strctre represented by the graph We explore here their basic mathematical properties present explicit formlae for these new graph invariants nder several graph operations Key words: Topological Index, Zagreb Indices Zagreb Coindices, raph Operations Cite this article: Zagreb Indices Zagreb Coindices of Some raph Operations International Jornal of Advanced Research in Engineering Technology, 7(3, 06, pp INTRODUCTION A representation of an object giving information only abot the nmber of elements composing it their connectivity is named as topological representation of an object A topological representation of a molecle is called moleclar graph A moleclar graph is a collection of points representing the atoms in the molecle set of lines representing the covalent bonds These points are named vertices the lines are named edges in graph theory langage The advantage of topological indices is in that they may be sed directly as simple nmerical descriptors in a comparison with physical, chemical or biological parameters of molecles in Qantitative Strctre Property Relationships (QSPR in Qantitative Strctre Activity Relationships (QSAR 5 editor@iaemecom

2 The first second Zagreb indices [6] which are dependent on the degrees of adjacent vertices appeared in the topological formla for the π-electron energy of conjgated systems, were first introdced by tman Trinajstic (983 The Zagreb indices are fond to have applications in QSPR QSAR stdies Recently, the first second Zagreb coindices [], a new pair of invariants, were introdced in Doslic which are dependent on the degrees of non-adjacent vertices thereby qantifying a possible inflence of remote pairs of vertices to the molecle s properties Note that the Zagreb coindices of are not agreb indices of hile the defining sms are over the set of edges of, the degrees are still with respect to ZAREB INDICES AND ZARED COINDICES All graphs considered are finite simple Let be a finite simple graph on n vertices m edges e denote the verte set the edge set by respectively oslic he complement of denoted by, is a simple graph on the same set of vertices V( in which two vertices v are adjacent (ie connected by an edge v, iff they are not adjacent in Hence v E( v E( Obviosly, E( E( E( K n n m E( m The degree of a vertex in is denoted by d( the degree of the same verte in is then given by d ( n d( The sm of the degrees of vertices of a graph is twice the nmber of edges (ie d( m V ( ZAREB INDICES The First Zagreb Index of [5] is denoted by M( is defined as M ( d( V ( The first Zagreb index can also be expressed as a sm over edges in M ( d ( d ( v ve ( The Second Zagreb Index of is denoted by M( is defined as M( d( d( v ve ( 6 editor@iaemecom

3 Zagreb Indices Zagreb Coindices of Some raph Operations Example Chemical compond: Methylcyclopropane Here 3 4 Figre v, v, v, v represents the labeling of the graph, while the nderlined nmbers st for the valencies of the vertices in M( 3 8 M( ZAREB COINDICES The First Zagreb Coindex[]of is denoted by M( is defined as M d d v ve ( ( ( ( The Second Zagreb Coindex of is denoted by M( is defined as 7 editor@iaemecom

4 M d d v ve ( ( ( ( EXAMPLE Chemical compond: Methylcyclopentane Figre Here E ( = {v v 5, v v 6, v v 3, v v 4, v v 5, v v 6, v 3 v 4, v 3 v 6, v 4 v 5 } M( = 34 M( = 3 3 BASIC PROPERTIES OF ZAREB COINDICES Proposition Let be a simple graph [4] on n vertices m edges Then M( M( ( n ( m m 8 editor@iaemecom

5 Zagreb Indices Zagreb Coindices of Some raph Operations Proof We know that M ( d ( M( d( V ( V ( = V ( n d ( n ( n d ( d ( V ( V ( V ( n n m n M ( 4 ( ( M( = ( n ( m m M( Proposition Let be a simple graph on n vertices m edges Then M( m( n M( Proof The First Zagreb Coindex of is n d ( n d ( v ve ( = M d d v ve ( ( ( ( ( n ( d ( d ( v ve ( m( n M ( n ( n m d ( V ( n ( n m ( n d( V ( nn ( ( n m ( ( ( ( n n d d V ( V ( V ( n n m n n n m n M ( ( ( 4 ( ( M ( m( n M ( Proposition 3 Let be a simple graph on n vertices m edges Then M ( M ( ( n M ( m( n 9 editor@iaemecom

6 Proof The Second Zagreb Coindex is M ( d( d( v ve ( ( n d ( ( n d ( v ve ( ( n ( n d ( d ( v d ( d ( v ve ( ve ( ve ( M( M ( ( n M ( m( n Proposition 4 Let be a simple graph on n vertices m edges Then M( M ( Proof By applying proposition to, we get M ( m( n M ( Now, plgging in the expression for M( from proposition yields M ( m( n M ( ( n ( m m m( n M ( M ( M ( Proposition 5 Let be a simple graph on n vertices m edges Then M( m M( M( Proof The reslt follows by sqaring both sides of the identity V ( d( m then Splitting the term d ( d ( v into two sms, one over the edges of the other over the edges of (tman 004 V ( ve ( ve ( d( d( 4 m d( d( d( ve ( V ( ve ( ve ( d ( d ( d ( d ( d ( 4m d d m d d d ( ( ( ( ( V ( ve ( 30 editor@iaemecom

7 Zagreb Indices Zagreb Coindices of Some raph Operations M( m M( M( 4 MAIN RESULTS In this section weintrodce the Zagreb indices coindicesof Cartesian prodct, Disjnction, Composition, Tensor prodct Normal prodct of graphs All considered operations are binary Hence, we sally deal with two finite simple graphs, For a given graph i, its vertex set edge set will be denoted by V i E i respectively, where i =, The nmber of edges in i is denoted by mi When more than two graphs can be combined sing a given operation, the vales of sbscripts will vary accordingly 4 CARTESIAN PRODUCT For given graphs, (Ashrafi 009 Cartesian prodct of is the graph V on the vertex sets V (, v (, v v are adjacent iff is v adjacent to or is adjacent to v v (, v The degree of a vertex of is given by d (, v d ( d ( v 4 DISJUNCTION The disjnction of two graphs is the graph [] with the vertex sets VV (, v (, v are adjacent iff is adjacent to or v is adjacent to v The degree of a vertex (, v of is given by d (, v nd ( nd ( v d ( d ( v 43 COMPOSITION The composition of two graphs is the graph with the vertex sets VV (, v (, v are adjacent iff [ is adjacent to ] or [ v v is adjacent to ] The degree of a vertex (, v of is given by d (, v n d ( d ( v 44 TENSOR PRODUCT Let ( V, E ( V, E be twographs whose vertex sets are disjoint as also their edge sets (Parthasarathy 994 [7] The tensor prodct of graphs has as its vertex set V V (, v (, v are adjacent iff is 3 editor@iaemecom

8 adjacent to v is adjacent to v The degree of a vertex ( i, j d (, v d ( d ( v given by i j i j v of is 45 NORMAL PRODUCT The normal prodct (strong prodct or wreath prodct of two graphs (BaskarBabjee 0 has as its vertex set VV (, v are adjacent iff is adjacent to v v v is adjacent to v or or (, v is adjacent to v is adjacent to v The degree of a vertex (, is given byd ( i, v j d ( i d ( v j d ( i d ( v j v of All graphs are finite simple We denote the vertex edge sets of by V( E( respectively The nmber of vertices edges of by V( n E( m respectively It can be easily seen that V( V( V( V( V( V V ( ( LEMMA 6 If are two simple graphs then, E( V( E( V( E( E( V( E( V( E( E( E( E( V( E( V( E( E( E( E( E( E( E( V( E( V( E( PROOF The part (a-e are conseqence of definitions some wellknown reslts of the book of Imrich Klavzar [3] ZAREB TYPE INDICES FOR CARTESIAN PRODUCT OF RAPHS THEOREM 7 The First Zagreb index of Cartesian prodct of two graphs is M ( n M ( n M ( 8m m i j 3 editor@iaemecom

9 Zagreb Indices Zagreb Coindices of Some raph Operations PROOF M ( n n d (, i v j i j Using the definition of Cartesian prodct, we get n n d ( ( i d v j i j n n d ( ( ( ( i d v j d i d v j i j n n n n n n d i d v j d i d v j i j i j i j ( ( ( ( n M ( n M ( 8m m THEOREM 8 The First Zagreb coindex of cartesian prodct of two graphs is M( nn m ( n m( n 8 mm n M( n M( PROOF: By Lemma (6-a Theorem 7, M ( m( n M ( Using the property M ( E( ( V( M ( M ( E( V( M ( n m n m ( n n M ( ( n m n m ( n n 8 m m n M ( n M ( ( nn ( nm nm 8m m n m ( n M( n m ( n M( n n ( n m n m ( n m n m 8m m m n n m n n M ( n n m m n n M ( M( nn m ( n m( n 8 mm n M( n M( 33 editor@iaemecom

10 ZAREB TYPE INDICES FOR DISJUNCTION OF RAPHS THEOREM 9 The First Zagreb index of disjnction of two graphs is M ( ( n 4 m n M ( ( n 4 m n M ( 8 mm n n M ( M ( PROOF: Using the definition of disjnction, we get M ( n n d (, i v j i j n n nd ( i nd ( v ( ( j d i d v j i j n n nd ( i nd ( v ( ( j d i d v j i j n d ( n d ( v d ( d ( v i j i j n n n n n n n d i n d v j nn d i d v j i j i j i j ( ( ( ( n n n n d i d v j nd i d v j i j i j ( ( ( ( n n i j i j n d ( d ( v n M ( n M ( n n ( m ( m M ( M ( 3 3 n M ( ( m n ( m M ( M ( ( n 4 m n M ( ( n 4 m n M ( 8 mm n n M ( M ( THEOREM 0 The First Zagreb coindex of disjnction of two graphs is M ( m n ( n m n ( n 4 m m ( n n 3 3 n mn m M( n m n m M( M( M( PROOF: By Lemma (6-b Theorem 9, M ( m( n M ( Using the property M ( E( ( V( M ( 34 editor@iaemecom

11 Zagreb Indices Zagreb Coindices of Some raph Operations M ( E( V( M ( n m n m m m ( n n M ( n n m n n m 4mm n n mn m n 4mm 3 3 ( n 4 m n M ( ( n 4 m n M ( 8 mm n n M ( M ( 4mm mm n n n n m n n m n m n m (4 mn n m ( n M( (4 mn n m( n M( m ( n M ( m ( n M ( M ( m n ( n m n ( n 4 m m ( n n 3 3 n mn m M( n m n m M( M( M( ZAREB TYPE INDICES FOR COMPOSITION OF RAPHS THEOREM The First Zagreb index of composition of two graphs is 3 M ( n M ( n M ( 8n mm PROOF: Using the definition of Composition, we get M ( n n d ( i, v j i j n n nd ( ( i d v j i j n n nd ( ( i d v j nd ( ( i d v j i j n n n n n n nd i d v j nd i d v j i j i j i j ( ( ( ( 3 M ( n M ( n M ( 8n mm THEOREM The First Zagreb coindex of composition of two graphs is 3 M ( n n m ( n n m ( n 8 n m m n M ( n M ( PROOF: By Lemma (6 c Theorem, 35 editor@iaemecom

12 M ( m( n M ( Using the property M ( E( ( V( M ( M ( E( V( M ( n m n m ( n n M ( 3 n n ( n m n m ( n m n m 8 n mm n M ( n M ( n n ( n m n m ( mn n m 8n mm 3 n m ( n M( n m( n M( From the above, after simple calclations, the desired reslts follows 3 M ( n n m ( n n m ( n 8 n m m n M ( n M ( ZAREB TYPE INDICES FOR TENSOR PRODUCT OF RAPHS THEOREM 3 If are two simple graphs if is connected then first Zagreb index of is M( M( M( PROOF: The first Zagreb index of is M ( n n d (, i v j i j n n d ( ( i d v j i j n ( d ( ( ( i d v (( d ( v j i n ( d ( i M( i M ( M ( M ( THEOREM 4 If are two simple graphs if is connected then second Zagreb index of is M( M( M( 36 editor@iaemecom

13 Zagreb Indices Zagreb Coindices of Some raph Operations PROOF The second Zagreb index of is M ( d (, v d (, v ( ( (, v (, v E ( (, v (, v E ( d ( d ( v d ( d ( v d ( d ( d ( v d ( v (, E ( (, E ( M ( M ( M ( THEOREM 5 If are two simple graphs if is connected then first Zagreb coindex of is M( 4 m m ( n n m( n M( m ( n M ( M ( M ( PROOF: By Lemma (6 d Theorem 3, M ( m( n M ( We know that M ( E( ( V( M ( M ( E( V( M ( E( E( V( V( M( M( 4m m n n M ( M ( 4m m n n m ( n M ( m ( n M ( 4 m m ( n n 4 m m ( n ( n m ( n M ( m ( n M ( M ( M ( 4 m m ( n n n n n n m ( n M ( m ( n M ( M ( M ( M ( 4 m m ( n n m ( n M ( m ( n M ( M ( M ( 37 editor@iaemecom

14 THEOREM 6 Let be two simple graphs then, is connected sch that second Zagrebcoindex of is M ( m M ( (m n m M ( (m n m M ( ( n m m M ( ( n m M ( M ( 4 m m ( m ( n m ( n ( n ( n M ( M ( M ( M ( M ( M ( PROOF: By Lemma (6 d, Theorem 3 Theorem 4, We know that M( m M( M( M( E( M( M( M( E( M( M( E( E( M( M( M( M( ( mm M( M( M( M( 8m m m M( M( m M( M( m ( n M ( ( ( m n M 8m m m M( m ( n M( m M( m( n M( m ( n M ( ( ( m n M By above calclations, we conclde that M ( m M ( (m n m M ( (m n m M ( ( n m m M ( ( n m M ( M ( 38 editor@iaemecom

15 Zagreb Indices Zagreb Coindices of Some raph Operations 4 m m ( m ( n m ( n ( n ( n M ( M ( M ( M ( M ( M ( ZAREB TYPE INDICES FOR NORMAL PRODUCT OF RAPHS THEOREM 7 The First Zagreb index of normal prodct of two graphs is M ( (4 m n M ( (4 m n M ( 8 m m M ( M ( PROOF n n i j i j i j M ( d ( d ( v d ( d ( v n n d ( ( ( ( i d v j d i d v j i j d ( d ( v d ( d ( v i j i j n n d ( ( ( ( i d v j d i d v j i j d ( d ( v d ( d ( v d ( d ( v i j i j i j n n n n n n d i d v j d i d v j i j i j i j ( ( ( ( n n n n d i d v j d i d v j i j i j ( ( ( ( n n d i d v j i j ( ( n M ( n M ( ( m ( m M ( M ( M ( ( m ( m M ( nm( nm ( 8 mm M( M( mm mm 4 ( 4 ( M ( (4 m n M ( (4 m n M ( 8 m m M ( M ( THEOREM 8 The First Zagreb coindex of normal prodct of two graphs is M ( n n m ( n n n m ( n 4 m m ( n n (4 m n M ( M ( M ( M ( ( n m n m M ( ( n m n m 39 editor@iaemecom

16 PROOF: By Lemma (6 e Theorem 7, M ( m( n M ( Using the property M ( E( ( V( M ( M ( E( V( M ( m m n m n m ( n n M ( M ( 4mm n n 4mm n n m n m n n m nm (4 m n M ( (4 m n M ( 8 m m M ( M ( 4mm n n 4mm n n m n m n n m nm (4 m n m( n M( (4 m n m ( n M( 8mm m ( n M ( m ( n M ( 4mm n n 4mm n n m n m n n m nm 8 mm ( n 4 m M( nm ( n n M mm n m M ( 8 ( 4 ( nm ( n n M mm mm n n ( 8 4 ( ( m ( n M ( M ( M ( M ( n n m ( n n n m ( n 4 m m ( n n (4 m n M ( M ( M ( M ( ( n m n m M ( ( n m n m CONCLUSION In this paper, we have stdied the Zagreb indices Zagreb coindices also we have investigated their basic mathematical properties obtained explicit formlae for compting their vales nder several graph operations namely Cartesian prodct, Disjnction, Composition, Tensor prodct Normal prodct of graphs Nevertheless, there are still many other graph operations special classes of graphs that are not covered here we mention some possible directions for ftre research 40 editor@iaemecom

17 Zagreb Indices Zagreb Coindices of Some raph Operations REFERENCES [] Ashrafi A R, T Doslic, A Hamzeb, The Zagreb coindices of raph operations, Discrete Applied Mathematics 58 ( [] BaskarBabjee J, Ramakrishnan S, Zagreb indices coindices for compond graphs, proceedings of ICMCM 0, PS Tech 9- Dec 0, Narosa pblications, pp: [3] W Imrich, S Klavzar, Prodct raphs: Strdtre Recognition, John Wiley & Sons, New York, USA 000 [4] Das K Ch, I tman, Some properties of the second Zagreb Index, MATCH Commn Math Compt Chem 5 ( [5] Khalifeh M H, H Yosefi-Azari, A R Ashrafi, The First Second Zagreb indices of some graph operations, Discrete Applied Mathematics 57 ( [6] Nenad Trinajstic, Chemical raph Theory, Volme II, CRC Press, Inc, Boca Raton, Florida, 983 [7] Parthasarathy K R, Basic raph Theory, Tata Mcraw Hill Pblishing Company Limited, New Delhi, editor@iaemecom