On Graphs with Same Distance Distribution

Transcription

1 Appled Mathematcs, 07, 8, ISSN Onlne: ISSN Prnt: On Graphs wth Same Dstance Dstrbuton Xulang Qu, Xaofeng Guo,3 Chengy Unversty College, Jme Unversty, Xamen, Chna School of Mathematcal Scences, Xamen Unversty, Xamen, Chna 3 College of Mathematcs and System Scences, Xnjang Unversty, Urumch, Chna How to cte ths paper: Qu, XL and Guo, XF (07) On Graphs wth Same Dstance Dstrbuton Appled Mathematcs, 8, Receved: Aprl 8, 07 Accepted: June 6, 07 Publshed: June 9, 07 Copyrght 07 by authors and Scentfc Research Publshng Inc Ths wor s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 40) Open Access Abstract In the present paper we nvestgate the relatonshp between Wener number W, hyper-wener number R, Wener vectors WV, hyper-wener vectors HWV, Wener polynomal H, hyper-wener polynomal HH and dstance dstrbuton DD of a (molecular) graph It s shown that for connected graphs G and G, the followng fve statements are equvalent: DD G DD G WV G WV G HWV G = HWV G, 4) ) ( ) = ( ), ) ( ) = ( ), 3) ( ) ( ) H( G) = H( G ), 5) HH ( G) = HH ( G ) ; and f G and G have same dstance dstrbuton DD then they have same W and R but the contrary s not true Therefore, we further nvestgate the graphs wth same dstance dstrbuton Some constructon methods for fndng graphs wth same dstance dstrbuton are gven Keywords Dstance Dstrbuton, Dstance Matrx, Wener Vector, Hyper-Wener Vector Introducton The Wener ndex s one of the oldest topologcal ndces of molecular structures It was put forward by the physco-chemst Harold Wener [] n 947 The Wener ndex of a connected graph G s defned as the sum of dstances between all pars of vertces n G : W= W( G) = dg ( uv, ) { uv, } V( G) G uv s the dstance between vertces u and v n G As an extenson of the Wener ndex of a tree, Randć [] ntroduced Wener where V ( G ) s the vertex set of G, and d (, ) matrx W and hyper-wener ndex R of a tree For any two vertces, j n DOI: 0436/am June 9, 07

2 X L Qu, X F Guo T, let (, π denote the unque path n T wth end vertces, j and the T π denote the components of T E( π (, ) length d j, let T, π (, and, (, contanng and j, respectvely, and let n, π (, and n, π (, denote the numbers of the vertces n T, π (, and T, π (,, respectvely Then the Wener W =, matrx W and the hyper-wener number R of T can be gven by ( w j ) wj = n, (, ) n π j, π(,, and R = w < j j In Refs [3] [4], Randc and Guo and colleagues further ntroduced the hgher Wener numbers and some other Wener matrx nvarants of a tree T The hgher Wener numbers can be represented by a Wener number sequence W, W, W,, where 3 R = W =,, W = w It s not dffcult to see W = W, and d, j=, < j j After the hyper-wener ndex of a tree was ntroduced, many publcatons [5]-[] have appeared on calculaton and generalzaton of the hyper-wener ndex Klen et al [5] generalzed the hyper-wener ndex so as to be applcable to any connected structure Ther formula for the hyper-wener ndex R of a graph G s [] R= RG ( ) = ( dg ( uv, ) + dg ( uv, )) { uv, } V( G) The relaton between Hyper-Wener and Wener ndex was gven by Gutman The Hosoya polynomal H( G ) of a connected graph G was ntroduced by Hosoya [] n 988, whch he named as the Wener polynomal of a graph: H H( Gx, ) d( G, ) x, = = where d( G, ) s the number of pars of vertces n the graph G that are dstance apart 0 In Ref [3], Cash ntroduced a new hyper-hosoya polynomal ( ) 0 ( + ) ( ) HH = HH G, x = d G, x The relatonshp between the Hosoya polynomal and the Hyper-Hosoya polynomal was dscussed [3] The sequence ( d( G, ), d( G, ), ) s also nown (snce 98) as the dstance dstrbuton of a graph G [4], denoted by DD ( G ) It s easy to see that W= d( G, ) 0 Later the defnton of hgher Wener numbers s extended to be applcable to any connected structure by Guo et al [5] For a connected graph G wth n vertces, denoted by,,, n, let wj, = max { dj +, 0} where d j s the dstance between vertces and j Then W = w < j j,, =,,, are W, W, s called the called the hgher Wener numbers of G The vector ( ) hyper-wener vector of G, denoted by HWV ( G ) The concept of the Wener vector of a graph s also ntroduced n ref [5] For a connected graph G wth n vertces, denoted by,,, n, let W,, = d < j, d( = j, =,, The vector WV G ( W W ) s called the Wener vector of G, denoted by ( ) Moreover, a matrx sequence ( ) ( ) ( ) (,, 3, ) W W W, called the Wener matrx 800

3 X L Qu, X F Guo ( ) ( H) sequence, and ther sum W = W, called the hyper-wener matrx, =,, ( ) are ntroduced, where W = D s the dstance matrx A Wener polynomal sequence and a weghted hyper Wener polynomal of a graph are also ntroduced In ths paper, based on the results n ref [5], we study the relaton between Wener number W, hyper-wener number R, Wener vector WV, hyper- Wener vector HWV, Hosoya polynomal H, hyper-hosoya polynomal HH and dstance dstrbuton DD of a graph It s shown that for connected graphs G and G, the the contrary s not true Ths means that the dstance dstrbuton of a graph s an mportant topologcal ndex of molecular graphs Therefore, we further nvestgate the graphs wth same dstance dstrbuton It s shown that the graphs wth same vertex number, edge number, and dameter have same dstance dstrbuton Some constructon methods for fndng graphs wth same dstance dstrbuton are gven The Relaton between W, R, WV, HWV, H, HH, DD dam G denote the dameter of a graph G Theorem Let G and G be connected graphs Then the followng fve statements are equvalent: ) G and G have same dstance dstrbuton DD ; ) G and G have same Wener vector WV ; 3) G and G have same hyper-wener vector HWV ; 4) G and G have same Wener polynomal H ; 5) G and G have same hyper-wener polynomal HH Proof We shall show the equvalent statements by () () (3) (4) (5) () () () By the defntons of DD and WV, Let ( ) DD( G) = ( d( G, ), d( G, ),, d( G, dam( G) )), and ( ) = (,,, ( ) ) = ( (,), (,),, ( ) (, dam G ( ))) Clearly, f DD ( G) = DD ( G ), then WV ( G) = WV ( G ) () (3) If WV ( G) = WV ( G ), then W = d = W = j d j WV G W W W d G d G dam G d G dam G for =,,, dam( G) So max {, 0} max {, 0 < j < j < j } < j for,,, dam( G) HWV G = HWV G < jd, j = < jd, j = W= w = d + = d + = w = W j, j j j, = Hence ( ) ( ) (3) (4) Suppose HWV ( G) = HWV ( G ) Then W W and dam( G) = dam( G ) If = dam( G) = dam( G ), then W = max { dj +, 0 } = d ( G, dam( G) < j ) = W = max { d, 0 } (, ( j j + = d G dam G < )) Assume, for < l dam( G), d( G, ) d( G, ) l l W = max{ dj l+, 0 } = d( G, l ) + max{ dj l+, 0} d( G, l ) d( G, )( l ) < j < j, dj > l = + + < j, dj = > l = for =,,, = Let = Then, and 80

4 X L Qu, X F Guo max {, 0 } (, ) max {, 0 < j j < j, d } j > l j l (, ) (, )( ) W = d l + = d G l + d l + l = d G l + d G l + = W By nducton hypothess, < jd, j = > l d ( G, )( l + ) = d ( G, )( l + ) So we have < j, dj = > l < j, dj = > l (, ) = (, ) Now t follows that d( G, ) = d( G, ) for =,,, and so H ( G, x ) (, ) ( = d G x = d G, ) x = H ( G, x ) d G l d G l 0 0 (4) (5) By the defntons of Hosoya polynomal H and hyper-hosoya polynomal HH, t s easy to see that, f H( G, x) H( G, x) HH ( G, x) = HH ( G, x) (5) () If HH ( G, x) = HH ( G, x), then d( G, ) d( G, ) =,, Therefore DD ( G) = DD ( G ) Theorem Let G and trbuton Then G and =, then = for G be two graphs wth same dstance ds- G have same W and R Proof: By the defntons of DD, W and R, ( ) ( ) = (,), (,),, (, ( )), W ( G ) = { } ( ) d ( uv, G ) DD G d G d G d G dam G and RG ( ) = { } ( ) d ( uv, ) + d ( uv, ), uv, V G uv, V G ( G G ) Clearly, f DD ( G) = DD ( G ), then W ( G) = W ( G ) and R( G) R( G ) = However, the contrary of the theorem doesn t hold For nstance, the trees T and T (resp T and T ) n Fgure have same W and R, but they have dfferent dstance dstrbutons 3 Graphs wth Same Dstance Dstrbuton From the above theorems, one can see that, f two graphs G and G have Fgure W( T) = W( T ) = 86, RT ( ) RT ( ) 66 DD( T ) = ( 8,3,9,5,) W( T) = W( T ) = 98, RT ( ) RT ( ) DD T =, DD( T ) = ( ) ( ) ( ) 8,0,8,5, 4, = =, ( ) ( ) 8,,6,5,6 DD T = 8,4,6,8, = = 7, 80

5 X L Qu, X F Guo same dstance dstrbuton DD, then they have same W, WW, WV, HWV, H and HH So t s sgnfcant to study the graphs wth same dstance ds- trbuton ) The mnmum non-somorphc acyclc graphs wth same DD By drect calculaton, we now the mnmum non-somorphc acyclc graphs wth same dstance dstrbuton are the followng two pars of trees n Fgure whch have 9 vertces ) The mnmum non-somorphc cyclc graphs wth same DD The mnmum non-somorphc cyclc graphs wth same dstance dstrbuton are the followng graphs wth 4 vertces (see Fgure 3) Note that, for the above graphs wth same dstance dstrbuton, ther Wener matrx sequences and hyper-wener matrces are dfferent The followng theorem gves a class of graphs wth same dstance dstrbuton Let be the set of all the graphs wth n vertces and m edges Theorem 3 Let GG,, and dam( G) = dam( G ) = Then DD ( G) = DD ( G ) Proof Snce dam G = dam G =, we have GG, and ( ) ( ) Fgure W( T3) = W( T3 ) = 8, RT ( 3) = RT ( 3 ) = 49, DD( T3) DD( T3 ) ( 8,3,,3) W( T4) = W( T4 ) = 9, RT ( 4) = RT ( 4 ) = 88, DD( T4) = DD( T4 ) = ( 8,0,0,6, ) = = Fgure 3 W( G) = W( G ) = 8, RG ( ) RG ( ) DD( G ) = DD( G ) = ( ) 4, = =, 0 803

6 X L Qu, X F Guo (,) = (,) =, (,) (,) d G d G m, and so DD ( G) DD ( G ) n d G d G = = m for 3 = n n Corollary 3 If > m> n+ dstance dstrbuton n n Proof For G wth > m> n+, =, = 0, d( G ) d( G ), then all graphs n have same, clearly dam( G) We assert that dam( G ) = Otherwse, there exst two vertces uv, V( G) such that d( uv, ) 3 Let P be a shortest ( uv, ) -path Then any vertex not on P s not adjacent to at least one of u and v, and the number of pars of non-adjacent vertces on P s equal to V ( P) + V P = V P V P So ( ) ( ( ) ) ( ( ) )( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) n m n V P V P V P, contradctng that n n = n ( V ( P) ) ( V ( P) 3) 4 n+ n m> n+ n Hence, by Theorem 3, f m> n+, all graphs n have same dstance dstrbuton Let G V H denote the graph obtaned from vertex-dsjont graphs G and H by connectng every vertex of G to every vertex of H Corollary 33 Let G, G n, m and G, G n, m Then G V G and G G have same dstance dstrbuton V V V, Proof Obvously, ( ) ( ) ( V ) = ( V ) =, and ( ) = ( ) = + + ( ) = ( ) dam G G dam G G V G G = V G G = n + n V V By Theorem 3, V V E G G E G G m m n n DD G G DD G G For graphs wth dameter greater than or equal to, we wll gve some constructon methods for fndng graphs wth same dstance dstrbuton Let G be a connected graph wth vertces set { v, v,, vn}, and let G D ( G) = ( d j ) be the dstant matrx of the graph G Let d ( v ) denote the number of the vertces at dstance from a vertex v n G, and let ( dam G ) G ( ) ( ), G ( ),, G ( ) ( ) DDG v = d v d v d v ) be the dstance dstrbuton of v n G Theorem 34 Let G and G (resp G and G ) be the connected graphs wth n (resp n ) vertces and wth same dstance dstrbuton For v V G v V G v V G v V G, let G (resp G ) be ( ), ( ), ( ), and ( ) the graph ob- taned from G and G (resp G and G ) by dentfyng v and v (resp v and v ) If DDG ( v ) ( ) = DDG v and DDG ( ) ( ) v = DDG v, then G and G have same dstance dstrbuton d G, = d G, for =,, Proof It s enough to prove ( ) ( ) 804

7 X L Qu, X F Guo Clearly, d ( G, ) d ( G, ) d G G ( G, ) d ( v ) d ( v ) = + + Smlarly, d G, = d G, + d G, + d v d v Because, j, + j= j G G, j, + j= j ( ) ( ) ( ) ( ) ( ) ( ) = DD ( G), DD ( G ) = DD ( G ), DDG ( v ) ( ) DDG v ( ) ( ) d G, = d G, for =,, DD G =, DDG v = DDG v, we have ( ) ( ) Hence DD ( G) = DD ( G ) Theorem 35 Let G, =,, and let S V ( G) such that any two vertces n S have dstance less than or equal to n G, and S = S Let G{ S } denote the graph obtaned from G by contractng vertces n S to a vertex s Let G be the graph obtaned from G by addng a new vertex x and connectng x to every vertex n S If DD ( G) = DD ( G) and DD { } ( s G ) { } ( ) S = DD s G S, then DD ( G ) = DD ( G) Proof Clearly, by the condtons of the theorem, { ( ) ( ) ( ) ( ) } { (, G S ( ), G S DD G } ( ), = DD G + DD x G = DD G + S + d x + d x ), =, So, f DD ( G) = DD ( G) and DD ( G ) = DD ( G ) and DD s = DD s, then DD = DD { } ( ) { } ( ) G S G S G G From Theorem 34, we have the followng corollary: Corollary 36 Let G, G and DD ( G) = DD ( G) Let H be a connected graph vertex-dsjont wth G and G For v V ( G), v V ( G), and u V ( H), let G (resp G ) be the graph obtaned from G (resp G ) and H by dentfyng v and u (resp v and u ) If DDG ( v ) ( ) = DDG v, then G and G have same dstance dstrbuton From Theorem 35, one can obtan graphs wth same dstance dstrbuton n from graphs n n, m s wth same dstance dstrbuton by addng a new vertex and some edges Fgure 4 shows two pars of graphs wth 5 vertces and 5 edges and wth same DD, one of whch has dameter and the other has dameter 3 Fgure 5 shows three pars of graphs wth 6 vertces and 6 edges and wth Fgure 4 W( G) = W( G) = 5, RG ( ) RG ( ) DD( G) = DD( G) = ( 5,5) W( G3) = W( G3) = 6, RG ( 3) = RG ( 3) = 3, DD( G3) DD( G3) ( 5,4,) = =, 0 = = 805

8 X L Qu, X F Guo Fgure 5 W( G4) = W( G4) = 6, RG ( ) RG ( ) DD( G4) = DD( G4) = ( 6,7,) W( G ) W( G ) RG ( 5) = RG ( 5) = 4, DD( G5) DD( G5) ( 6,6,3) W( G6) = W( G6) = 9, RG ( 6) = RG ( 6) = 49, DD( G ) = DD( G ) = ( ) 6 6 6,5,3, = =, = =, = = same DD, two of whch have dameter 3 and the other has dameter 4 It s easy to see that the graphs n Fgure 5 can be obtaned from graphs n Fgure 3, Fgure 4 by the constructon methods gven n Theorems 34, 35 However, the constructon methods are not complete There mght be some graphs wth same DD whch could not be obtaned by the above constructon methods Open Problem Is there a constructon method for fndng all graphs wth same dstance dstrbuton? Acnowledgements Ths wor s jontly supported by the Natural Scence Foundaton of Chna (087, , 3600), the Scentfc Research Fund of Fujan Provncal Educaton Department of Chna (JAT6069) References [] Wener, H (947) Structural Determnaton of Paraffn Bolng Ponts Journal of the Amercan Chemcal Socety, 69, [] Randć, M (993) Novel Molecular Descrptor for Structure-Property Studes Chemcal Physcs Letters,, [3] Randć, M, Guo, XF, Oxley, T and Krshnapryan, H (993) Wener Matrx: 806

9 X L Qu, X F Guo Source of Novel Graph Invarants Journal of Chemcal Informaton and Computer Scences, 33, [4] Randć, M, Guo, XF, Oxley, T, Krshnapryan, h and Naylor, L (994) Wener Matrx Invarants Journal of Chemcal Informaton and Computer Scences, 34, [5] Klen, DJ and Gutman, I (995) On the Defnton of the Hyper-Wener Index for Cycle-Contanng Structures Journal of Chemcal Informaton and Computer Scences, 35, [6] Luovts, I and Lnert, W (995) A Novel Defnton of the Hyper-Wener Index for Cycles Journal of Chemcal Informaton and Computer Scences, 34, [7] Klen, DJ and Randć, M (993) Resstance Dstance Journal of Mathematcal Chemstry,, [8] L, XH (003) The Extended Hyper-Wener Index Canadan Journal of Chemstry, 8, [9] Klavzar, S, Gutman, I and Mohar, B (995) Labelng of Benzenod Systems whch Reflects the Vertex-Dstance Relatons Journal of Chemcal Informaton and Computer Scences, 35, [0] Cash, GG, Klavzar, S and Petovse, M (00) Three Methods for Calculaton of the Hyper-Wener Index of Molecular Graphs Journal of Chemcal Informaton and Computer Scences, 4, [] Gutman, I (00) Relaton between Hyper-Wener and Wener Index Chemcal Physcs Letters, 364, [] Hosoya, H (998) On Some Countng Polynomnals n Chemstry Dscrete Appled Mathematcs, 9, [3] Cash, GG (00) Relatonshp between the Hosoya Polynomnal and the Hyper-Wener Index Appled Mathematcs Letters, 5, [4] Bucley, F and Supervlle, L (98) Dstance Dstrbutons and Mean Dstance Problem Proceedngs of Thrd Carbbean Conference on Combnatorcs and Computng, Unversty of the West Indes, Barbados, January 98, [5] Guo, XF, len, DJ, Yan, WG and Yeh, Y-N (006) Hyper-Wener Vector, Wener Matrx Sequence, and Wener Polynomnal Sequence of a Graph Internatonal Journal of Quantum Chemstry, 06,

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