A Result on a Cyclic Polynomials
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1 Gen. Math. Note, Vol. 6, No., Feruary 05, pp ISSN 9-78 Copyrght ICSRS Pulcaton, 05.-cr.org Avalale free onlne at A Reult on a Cyclc Polynomal S.A. Wahd Department of Mathematc & Stattc U.W.I, Trndad, Wet Inde E-mal: hanaz@hotmal.com (Receved: -9- Accepted: -- Atract Th paper etalhe a reult on matchng polynomal that related to a conecture y Gutman, ee []. The prncple of Incluon and Excluon ued to count matchng of certan reduced ugraph. A functon then defned on each et of matchng to otan a reult on acyclc polynomal. Keyord: Matchng, Acyclc polynomal, Weght, Path and Cover. Introducton In the materal hch follo e conder fnte undrected graph thout loop and multple edge. Let G e uch a graph th p node. By a matchng n G, e mean a pannng ugraph hoe component are node and edge only. A - matchng a matchng th edge. Let u agn eght and to each node and edge repectvely n G. If the numer of -matchng n G then the total eght of the -matchng G. The matchng polynomal of G, ee Farrell [, ] defned a
2 60 S.A. Wahd p M (G a 0 p The acyclc polynomal a defned y Gutman [5]., α (G x p p a x ( 0. Th polynomal ealy otaned from matchng polynomal y replacng y x and y. For further relaton eteen the to polynomal, ee Farrell []. Gutman conecture a follo: Let G e a graph and A, B are to ugraph of G uch that V(A V(B φ. Let P,P,..., P e the path n G hoe one endvertex elong to A and the other endvertex elong to B and no other node elong to ether A or B. Then α (G α (G - A B α (G - A α (G - B - α (G - A P α (G - B P α (G - A P P α (G - B P P... < ( α (G - A P P... P α (G - B P P... P Where the conventon that henever at leat to among the path P, P,...,P have at leat one common vertex, then α ( G - A - P P... P α (G - B - P P... P 0. Gutman conecture cloely related to the theory of Jaco polynomal. Smlar (ut not equvalent reult ere earler pulhed n Helmann and Le (6 and Godl []. In th paper a reult gven hch conder the cae hen the graph A and B may have common node. A a conequence, Gutman conecture ould follo. A path a tree hch ha exactly to endnode. The graph G-A otaned y removng the node of A.e. V(A and all the ncdent edge from G. We ometme rte α ( G -A x a α (G A.
3 A Reult on a Cyclc Polynomal 6 The Man Reult Let G e a laeled graph and A, B are to ugraph. Let P e the path a defned aove. The length of P It convenent to rte M(G, a M(G. Then M (G (A B M (G -( A B M (G - A M (G - B - M (G - A P M (G - B P M (G - A P P < M (G - (- B P P ( ( M (G - A P... P M (G - B P... P... (-... here y conventon M(φ. Proof: In th proof e frt ue the Prncple of Incluon and Excluon to dentfy a numer of matchng and then e apply a functon f to convert the matchng nto matchng polynomal. Let A and B e laeled ugraph of G and P, P,..., P e the path a defned aove. To graph G A - P and G B - P are contructed from G th repect to a path P th no lael repeated. We let e the property that a matchng otaned from the ugraph G A - P and G B - P for a path P. In th ay the numer of matchng decred th repect to property Smlarly properte... and. the numer of matchng decred th repect to. Ung the prncple of ncluon and excluon, e get N <... ( ( Frtly e examne the term on the rght de of equaton. We convert a matchng to t matchng polynomal y ung the functon f a follo: f defned a f ( M(G A P M(G B P ( here the length of path P.
4 6 S.A. Wahd N the numer of matchng of the graph ( G A (G B. In th cae no path are removed and thu n the unretrcted cae only graph A and B are removed once and eparately from G. On applyng f e get M(G ( 0 A M(G B nce no path condered. the numer of matchng of G A P P and G B P P. Applyng f e get M(G A P P M(G B P P ( nce the path P and P have and edge repectvely. All hgher ordered term on the rght hand de of equaton ( are found a decred. The term on the left hand de of ( no analyzed. We mut fnd the to ugraph that are to e removed from G. In examnng the complment property, the follong mut e noted: (a In fndng graph A and B are removed once and eparately. Thu n thee graph are removed once ut not eparately. ( In fndng one endnode of path P removed from G A and the other endnode from G B. Alo the entre path P removed from G A and G B. Thu n fndng N (, the entre path P not to e removed. In addton, n fndng N ( oth endnode of path P are trvally removed nce the graph A and B are removed. Thu the hole of P not to e removed. We no examne ho the endnode of P are to e removed. (c In fndng oth endnode of N ( P are removed eparately. Thu n fndng, the par of endnode are removed together. Alo not removng one end node eparate from the other the ame a removng zero endnode together from ome graph X and oth endnode together from another graph Y. In conderng..., the pont (a, ( and (c aove are veed th repect to the complment of all properte. We need to dentfy the graph X and Y a e are conderng matchng of G X and G Y. The graph G (A B and G (A B atfy all the properte aove. A and B are removed once n the term G (A B and not eparately th repect to argument (a. Recall that
5 A Reult on a Cyclc Polynomal 6 a path P elected uch that there one endnode n A and the other n B. There are no path P that have an endnoden A B and next endnode n A B. The removal of ugraph A B ould enure that no entre path are removed a tated n argument (. Th the ame a removng zero endnode trvally from ome graph and removng oth endnode together from another graph. Removng A B from G enure that oth endnode of each path together. Thu X A B and Y A B. On collectng all term e get, M(G (A BM(G (A B M(G AM(G B M(G A P M(G B P ( M(G A P < ( M(G A P P M(G B P P (... P M (G B P S J... P ( S... P are removed... here y conventon M ( φ. ( In order to convert ( nto a reult on acyclc polynomal e ue the converon a tated n the ntroducton. For dont ugraph A and B, equaton( Gutman conecture. We llutrate th an example here A B Example φ. Let V(A {,,, 6} and V(B {,, 5} and G the follong :
6 6 S.A. Wahd The path are P {5,6}, P {5,} and P {, }. The follong calculaton can e ealy confrmed. M(G M(G A (A B M(G B, M(G (A B M (G A P M(G B P M(G A P M (G B P M(G A P M(G B P M (G A P P M(G B P P M(G A P P M(G B P P. 6 M(G (A BM(G (A B ( 6 8. On R.H.S. e have ( ( ( ( ( 6 ( ( ( 6. ( 8 (. ( ( Th converted to acyclc polynomal a 6 x x 7 x 6x 8x x x x x x. 5 6 x 8x x. x x x x Reference [] E.J. Farrell, An ntroducton to matchng polynomal, J. Com. Theory (B, 7(979, [] E.J. Farrell, The matchng polynomal and t relaton to the acyclc polynomal of a graph, Ar Comnatora, 9(980, -8. [] C.D. Godl, Real graph polynomal, In: J.A. Bondy and U.S.R. Murty (Ed, Progre n Graph Theory, Academc Pre, Toronto, (98, 8-9. [] I. Gutman, Some relaton for graphc polynomal, Pulcaton de L Inttut Mathematque, Nouvelle Sere Tome, 9(5 (986, 55-6.
7 A Reult on a Cyclc Polynomal 65 [5] I. Gutman, The acyclc polynomal of a graph, Pul. Int. Math (Beograd, (6 (977, [6] O.J. Helmann and E.H. Le, Monomer and dmer, Phyc. Rev. Lett., (970, -.
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