Research Article The Terminal Hosoya Polynomial of Some Families of Composite Graphs

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1 Iteratioal Combiatorics, Article ID , 4 pages Research Article The Termial Hosoya Polyomial of Some Families of Composite Graphs Emeric Deutsch 1 ad Jua Alberto Rodríguez-Velázquez 2 1 Polytechic Istitute of New York Uiversity, Brookly, NY 11201, USA 2 Departamet d Egiyeria Iformàtica i Matemàtiques, Uiversitat Rovira i Virgili, Aveida Països Catalas 26, Tarragoa, Spai Correspodece should be addressed to Jua Alberto Rodríguez-Velázquez; juaalberto.rodriguez@urv.cat Received 27 December 2013; Accepted 31 March 2014; Published 16 April 2014 Academic Editor: Jiag Zeg Copyright 2014 E. Deutsch ad J. A. Rodríguez-Velázquez. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Let G be a coected graph ad let Ω(G) be the set of pedet vertices of G. Thetermial Hosoya polyomial of G is defied as TH(G, t) := x,y Ω(G):x =y t dg(x,y),whered G (x, y) deotes the distace betwee the pedet vertices x ad y.ithisotepaperwe obtai closed formulae for the termial Hosoya polyomial of rooted product graphs ad coroa product graphs. 1. Itroductio Throughout this paper G = (V, E) deotes a coected, simple, ad fiite graph with vertex set V=V(G)ad edge set E=E(G).Thedistace d G (u, V) betwee two vertices u ad V is the miimum of the legths of paths betwee u ad V. The Wieer idex W(G) of a graph G with vertex set {V 1, V 2,...,V }, defied as the sum of distaces betwee all pairsofverticesofg, W (G) := 1 2,j=1 d G (V i, V j ), (1) is the first mathematical ivariat reflectig the topological structure of a molecular graph. This topological idex has bee extesively studied; for istace, a comprehesive survey o the direct calculatio, applicatios, ad the relatio of the Wieer idex of trees with other parameters of graphs ca be foud i [1]. Moreover, a list of 120 refereces of the mai works o the Wieer idex of graphs ca be foud i the referred survey. The Hosoya polyomial of a graph was itroduced i Hosoya s semial paper [2] i 1988 ad received a lot of attetio afterwards. The polyomial was later idepedetly itroduced ad cosidered by Saga et al. [3] uder the ame Wieer polyomial of a graph. Both ames are still used for the polyomial but the term Hosoya polyomial is owadays used by the majority of researchers. The mai advatage of the Hosoya polyomial is that it cotais a wealth of iformatio about distace based graph ivariats. For istace, kowig the Hosoya polyomial of a graph, it is straightforward to determie the Wieer idex of a graph as the first derivative of the polyomial at the poit t = 1. Cash[4] oticed that the hyper-wieer idex ca be obtaied from the Hosoya polyomial i a similar simple maer. A comprehesive survey o the mai properties of the Hosoya polyomial, which icludes a wide list of bibliographic refereces, ca be foud i Gutma et al. [5]. Also,Estradaetal.[6] studied several chemical applicatios of the Hosoya polyomial. The Hosoya polyomial of a coected graph G is defied as H (G, t) := x,y V(G):x =y t d G(x,y). As we poited out above, the Wieer idex of a graph G is determied as the first derivative of the polyomial H(G, t) at t=1. (2)

2 2 Iteratioal Combiatorics The Hosoya polyomial has bee obtaied for trees, composite graphs, bezeoid graphs, tori, zig-zag opeeded aotubes, certai graph decoratios, armchair opeeded aotubes, zigzag polyhex aotorus, TUC 4 C 8 (S) aotubes, petachais, polypheyl chais, the circumcoroee series, Fiboacci ad Lucas cubes, Haoi graphs, ad so forth. See the refereces i [7]. The degree of a vertex V i G is the umber of edges icidet to it ad is deoted by deg G (V).Ifdeg G (V) =1,the V is called a pedet vertex of G. The set of pedet vertices of G will be deoted by Ω(G). Gutma et al. proposed the cocept of termial Wieer idex, which is defied as the sum of distaces betwee all pairs of pedet vertices of G [8]. For detailed iformatio o the termial Wieer idex the readers may refer to the recet papers [8 11] ad the refereces cited therei. By aalogy to the defiitio of Hosoya polyomial, the termial Hosoya polyomial was defied by Narayakar et al. [12]as TH (G, t) := x,y Ω(G):x =y t d G(x,y), ad recetly Ramae et al. studied TH(G, t) for the case of lie graphs [13]. As we ca expect, the termial Wieer idex, deoted by TW(G),isobtaiedfromTH(G, t) as (TH (G, t)) TW (G) =. (4) t t=1 Amog the most kow products of graphs, amely, Cartesia product, lexicographic product, strog product, rooted product, ad coroa product, the two latest oes are the oly cases i which we ca obtai coected graphs havig pedet vertices. So, i this paper we obtai formulae for the termial Hosoya polyomial of both types of graphs, rooted product graphs ad coroa product graphs. The remaiig defiitios will be give the first time that the cocept is foud i the text. 2. The Termial Hosoya Polyomial of Rooted Product Graphs A rooted graph is a graph i which oe vertex is labeled i a special way so as to distiguish it from the other vertices. The special vertex is called the root of the graph. Let G be a labeled graph o vertices. Let F be a sequece of rooted graphs G 1,G 2,...,G.Therooted product graph G(F) is the graph obtaied by idetifyig the root of G i with the ith vertex of G [14]. I this paper we cosider a particular case of rooted product graphs where F cosists of isomorphic rooted graphs [15]. More formally, assumig that V(G) = {u 1,...,u } ad that the root vertex of G is V, we defie the rooted product graph G V G asthegraphwhosevertexsetis V(G V G )=V(G) V(G ) ad whose edge set is E(G V G )= {(u i,b)(u i,y):by E(G )} {(u i, V)(u j, V) :u i u j E(G)}. (3) (5) Figure 1: The graph C 6 V P 3,whereC 6 is the cycle graph of order 6,P 3 is a path of order 3, adv is the cetral vertex of P 3. This chemical graph that correspods to the Cyclohexae (C6H12) is a aturally occurrig chemical that is also produced sythetically ad used as a solvet i umerous idustries. As a example of a chemical compoud whose graph is obtaied as a rooted product graph we cosider the Cycloalkaes with a sigle rig, whose chemical formula is C H 2, ad whose molecular graph ca be expressed as C V P 3,whereC is the cycle graph of order, P 3 is a path of order 3,adV is the cetral vertex of P 3 (see Figure 1). To begi with, we eed some additioal otatio. We defie the termial Hosoya polyomial with respect to u as TH u (G, t) := t dg(u,w). w Ω(G) {u} Theorem 1. Let G be a coected graph of order 2ad let G be a coected otrivial graph. The for ay V V(G ) Ω(G ), (6) TH (G V G,t) = TH (G,t) +H(G, t) (TH V (G,t)) 2. (7) Proof. Let V(G) = {u 1,u 2,...,u } ad let G i be the ith- copy of G i G V G ;thatis,g i is the subgraph of G VG geerated by the set {u i } V(G ), which is isomorphic to G.The, Ω(G V G )= Ω(G i ). (8) So, if Ω(G )=0,theΩ(G V G )=0ad, as a cosequece, TH V (G,t) = TH(G,t) = TH(G V G,t) = 0. So, we assume that Ω(G ) =0. Notice that for every pair x, y of pedet vertices of G ad u i V(G) we have d G V G ((u i,x),(u i,y))=d G i ((u i,x),(u i,y))=d G (x, y). (9) Also, for u i,u j V(G), i =j,wehave d G V G ((u i,x),(u j,y))= d G (x, V) +d G (u i,u j ) +d G (V,y). (10) Hece, TH(G V G,t) splits as the sum of two polyomials P(t) + Q(t), where P(t) is obtaied from pairs of pedet

3 Iteratioal Combiatorics 3 vertices i the same G i ad Q(t) is obtaied from pairs (u i, x), (u j,y),wherei=j.otheoehad, P (t) = ( x,y Ω(G ):x =y ad, o the other had, Q (t) = u i,u j V(G):i =j =(TH V (G,t)) 2 t d G (x,y) )= TH (G,t), (11) t d G(u i,u j ) ( ( y Ω(G ) =(TH V (G,t)) 2 H (G, t). Therefore, the result follows. x Ω(G ) t d G (y,v) ) t d G (x,v) ) t d G(u i,u j ) u i,u j V(G):i =j Give a vertex V V(G ), we defie d T (V) := d G (V,w). w Ω(G ) {V} (12) (13) Notice that we ca see d T (V) as the termial Wieer idex with respect to V; thatis,d T (V) is the first derivative of the polyomial TH V (G,t) at t = 1.Asadirectcosequece of Theorem 1 we obtai the followig result o the termial Wieer idex. Corollary 2. Let G be a coected otrivial graph. The for ay V V(G ) Ω(G ) adaycoectedgraphg of order 2, TW (G V G,t)= TW (G 2 )+W(G) Ω(G ) (14) +( 1) Ω(G ) d T (V). Now we cosider the case of rooted product graphs G V G where the root V is a pedet vertex of G. Theorem 3. Let G be a coected graph of order 2ad let G be a coected otrivial graph. The for ay V Ω(G ), TH (G V G,t)= (TH (G,t) TH V (G,t)) +H(G, t) (TH V (G,t)) 2. (15) Proof. We proceed as i the proof of Theorem 1,withtheoly differece that i this case P(t) is obtaied as follows: P (t) = =( ( x,y Ω(G ) {V}:x =y x,y Ω(G ):x =y t d G (x,y) ) t d G (x,y) =(TH (G,t) TH V (G,t)). x Ω(G ) {V} t d G (x,v) ) (16) Corollary 4. Let G be a coected otrivial graph. The for ay V Ω(G ) adaycoectedgraphg of order 2,oe has TW (G V G,t)= (TW (G ) d T (V)) +W(G) ( Ω(G ) 1)2 +( 1) ( Ω(G ) 1)d T (V). 3. The Termial Hosoya Polyomial of Coroa Graphs (17) Let G ad G be two graphs of order ad,respectively. The coroa product G G is defied as the graph obtaied from G ad G by takig oe copy of G ad copies of G ad joiig by a edge each vertex from the ith-copy of G with the ith-vertex of G [16]. We will deote by V={V 1, V 2,...,V } the set of vertices of G ad by G i =(V i,e i ) the copy of G such that V i is adjacet to V for every V V i.thus,thevertexsetof G G is V ( V i) ad, cosequetly, G G has order ( +1)ad, if G is coected, the G G is also coected ad it has diameter D(G G )=D(G)+2. Notice that the coroa graph K 1 G is isomorphic to the joi graph K 1 +G.Now,ifGis a tree ad G =N r is a empty graph(aemptygraphorodes cosists of r isolated odes with o edges. Such graphs are sometimes also called edgeless graphs or ull graphs), the the coroa graph G N r is a tree ad the set of leaves (pedet vertices) of G N r is V i.i particular, if G=Pis a path, the P N r is a caterpillar graph ad the path P, that is formed by the oleaves of P N r,is kow as the backboe of the caterpillar. The formula for the Hosoya polyomial of coroa graphs was previously obtaied i [17]. I this sectio we obtai the formula for the termial Hosoya polyomial of coroa graphs G G.Ofcourse,G G has pedet vertices if ad oly if G has isolated vertices. Corollary 5. For ay coected graph G of order 2ad ay graph G havig r isolated vertices, TH (G G,t)=( r 2 )t2 +r 2 t 2 H (G, t). (18) Proof. First of all, otice that G G =G V ( V +G ),where V isthetrivialgraphcomposedbythevertexv.now,sice TH( V +G,t)=( r 2 ) t 2 ad TH V ( V +G,t)=rt,theresult immediately follows from Theorem 1. Fially, Corollary 5 (ad also Corollary 2) leadstothe followig formula for the termial Wieer idex of G G. Remark 6. For ay coected graph G of order 2ad ay graph G havig r isolated vertices, TW (G G )=r(r 1) +r 2 W (G). (19)

4 4 Iteratioal Combiatorics Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Refereces [1] A. A. Dobryi, R. Etriger, ad I. Gutma, Wieer idex of trees: theory ad applicatios, Acta Applicadae Mathematicae. A Iteratioal Survey Joural o Applyig Mathematics ad Mathematical Applicatios,vol.66,o.3,pp ,2001. [2] H. Hosoya, O some coutig polyomials i chemistry, Discrete Applied Mathematics, vol. 19, o. 1 3, pp , [3] B.E.Saga,Y.Yeh,adP.Zhag, TheWieerpolyomialof agraph, Iteratioal Quatum Chemistry, vol.60, o. 5, pp , [4] G. G. Cash, Relatioship betwee the Hosoya polyomial ad the hyper-wieer idex, Applied Mathematics Letters, vol. 15, o.7,pp ,2002. [5] I. Gutma, Y. Zhag, M. Dehmer, ad A. Ilic, Alteburg, Wieer, ad Hosoya polyomials, i Distace i Molecular Graphs-Theory, I.GutmaadB.Furtula,Eds.,pp.49 70, Uiversity of Kragujevac, Kragujevac, Serbia, [6] E. Estrada, O. Ivaciuc, I. Gutma, A. Gutierrez, ad L. Rodríguez, Exteded Wieer idices. A ew set of descriptors for quatitative structure-property studies, New Chemistry, vol. 22, o. 8, pp , [7] E.DeutschadS.Klavžar, ComputigtheHosoyapolyomial of graphs from primary subgraphs, MATCH Commuicatios i Mathematical ad i Computer Chemistry,vol.70,o.2,pp , [8] I. Gutma, B. Furtula, ad M. Petrović, Termial Wieer idex, Mathematical Chemistry, vol. 46, o. 2, pp , [9] L.A.Székely, H. Wag, ad T. Wu, The sum of the distaces betweetheleavesofatreeadthe semi-regular property, Discrete Mathematics, vol. 311, o. 13, pp , [10] X. Deg ad J. Zhag, Equiseparability o termial Wieer idex, Applied Mathematics Letters,vol.25, o.3,pp , [11] Y.-H. Che ad X.-D. Zhag, O Wieer ad termial Wieer idices of trees, MATCH Commuicatios i Mathematical ad i Computer Chemistry,vol.70,o.2,pp ,2013. [12] K. P. Narayakar, S. S. Shirkol, H. S. Ramae, ad S. B. Lokesh, Termial Hosoya polyomial of thor graphs, i Iteratioal Coferece o Discrete Mathematics (ICDM 13), Karatak Uiversity, Dharwad, Idia, [13] H. S. Ramae, A. B. Gaagi, K. P. Narayakar, ad S. S. Shirkol, Termial Hosoya polyomial of lie graphs, Joural of Discrete Mathematics, vol. 2013, Article ID , 3 pages, [14] C.D.GodsiladB.D.McKay, Aewgraphproductadits spectrum, Bulleti of the Australia Mathematical Society,vol. 18,o.1,pp.21 28,1978. [15] A. J. Schwek, Computig the characteristic polyomial of a graph, i Graphs ad Combiatorics,R.BariadF.Harary,Eds., pp , Spriger, Berli, Germay, [16] R. Frucht ad F. Harary, O the coroa of two graphs, Aequatioes Mathematicae,vol.4,pp ,1970. [17] D. Stevaović, Hosoya polyomial of composite graphs, Discrete Mathematics,vol.235,o.1 3,pp ,2001.

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