JORGE LUIS AROCHA AND BERNARDO LLANO. Average atchig polyoial Cosider a siple graph G =(V E): Let M E a atchig of the graph G: If M is a atchig, the a

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1 MEAN VALUE FOR THE MATCHING AND DOMINATING POLYNOMIAL JORGE LUIS AROCHA AND BERNARDO LLANO Abstract. The ea value of the atchig polyoial is coputed i the faily of all labeled graphs with vertices. We dee the doiatig polyoial of a graph whose coeciets euerate the doiatig sets for a graph ad study soe properties of the polyoial. The ea value of this polyoial is deteried i a certai special faily of bipartite digraphs. 1. Itroductio The goal of this paper is to copute the average polyoials for the well-ow atchig polyoial ad the doiatig polyoial i certai classes of graphs. The atchig polyoial rst appeared i a paper by Heila ad Lieb [5] as a therodyaic partitio fuctio. For a very iterestig itroductio to its cobiatorial study as well as ay of its properties we refer the reader to [] ad [3]. The otio of doiatio i graphs was itroduced last cetury. This theory ca be cosulted i the boos by Ore [10] ad Berge [1]. The paper [7] shows recet developets of the theory ad a large accout of refereces o the topic. I the rst part of the paper we calculate the so called average atchig polyoial i the class of all labelled graphs with vertices. The subsequet sectios are devoted to the doiatig polyoial: deitio ad basic properties. Fially, we deterie the average doiatig polyoial i a certai class of bipartite digraphs. For the teriology of graph theory used here, see [9]. Date. May 9, Matheatics Subect Classicatio. Priary 05C70, 05A15. Key words ad phrases. atcchig, atchig polyoial, doiatig set. This paper is i al for ad o versio of it will be subitted for publicatio elsewhere. 1

2 JORGE LUIS AROCHA AND BERNARDO LLANO. Average atchig polyoial Cosider a siple graph G =(V E): Let M E a atchig of the graph G: If M is a atchig, the ay M 0 M is a atchig too. For V = we have that M = ad if the equality holds, the the atchig is called perfect. Let (G) deote the uber of atchigs of cardiality ( N) of a graph G ad by covetio, 0 (G) = 1: The atchig polyoial is deed by (G t) = [=] =0 (;1) (G) t ; : There are basic properties of the atchig polyoial studied i [3]. We recall soe of these properties that will be used later i this paper. Theore.1. (G t) = (G ; e t) ; (G ; i ; t) where i V ad e = fi g E: Applyig this theore to the coplete graph, we have the followig result. Theore.. For the coplete graph K (K t)=he (t) = ;= H t= p = where [=] =0 H (t) = (;1) d t e dx e;t (;1) ( ; ) t; ad He (t) = (;1) e t = d dx e;t = are the Herite ad the special Herite polyoial respectively. Further iforatio o Herite polyoials ca be foud i [8]. Let G the set of all labeled graphs with vertices. We dee (t) = ;( ) (G t) GG to be the ea value of the atchig polyoial i the set G or the average atchig polyoial i G : If G deotes the set of all labeled

3 MEAN VALUE 3 graphs with vertices ad edges, the we ca dee the average atchig polyoial i this set by (.1) (t) = ;1 It is ot dicult to establish that (.) Lea.3. (t) = (t) = ;( ) ( ) [=] (;1) =0 GG (G t) : ;1 (t) : (K ) t ; : Proof. Applyig the deitio of the atchig polyoial to (.1), we have that (t) = = [=] ;1 GG (;1) [=] ;1 (;1) (G) t ; 4 GG 3 (G) 5 t ; : We copute the su i bracets. Let M(G ) bethesetofatchigs of cardiality of G the (.3) (G) = 1 GG GG M M (G ) But ay atchig of a graph G G is a atchig of the coplete graph K so the su i the right of(.3) is equal to (.4) M M (K ) GG M M (G ) The secod su of (.4) is the uber of graphs belogig to G which cotai a xed atchig with exactly edges. Fixig this atchig, fro the other ; edges of K we ca choose the ; issig edges of G G i ; ; 1:

4 4 JORGE LUIS AROCHA AND BERNARDO LLANO ways. Therefore, the expressio (.4) is equal to ( ); as desired. (t) = Theore.4. = [=] [=] (;1) (;1) (t) = ; H (t) = ;1 [=] ;1 ; ; ; (K ) t ; (K ) t ; ; 1 (K ) t ; : (K ) ad Proof. Applyig lea 3.4 to the relatio (.), we obtai that (t) = ;( ) ( ) Sice the =0 = ;( ) [=] as was to be show. ( ) =0 [=] (;1) (t) = [=] (;1) ;1 ;1 (K ) t ; ( (K ) t ; 6 ) 4 = ( ); ; 1 (K ) t ; = : 3. The doiatig polyoial, defiitio ad properties Let G = (V E) be a siple graph ad D V: A set of vertices D is said to be a doiatig set if for every y V ; D there exists x D such that fx yg E: For ay vertex x V, let N (x) deote the eighbourhood of x, the set of all vertices adacet to x: We write N [x] =N (x) [fxg the closed eibourhood of x: With this otatio, D V is a doiatig set if for every y V ; D we have that N [y] \ D 6= : The faily of all doiatig sets of a graph G is deoted by D G : Observe that D G ad if D D G ad D D 0 the D 0 D G :

5 MEAN VALUE 5 For ay graph G the uber of doiatig sets of cardiality is deoted by (G) : We dee by (G t) = i=1 (G) t ; the doiatig polyoial of the graph G where = V : Sice D G the 0 (G) =0: For exaple, the doiatig polyoials of the coplete graph K ad the totally discoected graph K are (K t)= t ; =(1+t) ; t ad K t =1 =1 sice every subset of vertices of K is a doiatig set ad there is oly oe doiatig set of K : If deotes the epty graph, the ( t) = 0 sice D : Let S i=1 G i be a graph coposed of disoit subgraphs G 1, G,..., G : Theore 3.1. (G 1 [ G t)= (G 1 t) (G t) : Proof. There are o edges betwee V (G 1 )adv (G ) therefore D 1 V (G 1 ) ad D V (G ) are doiatig sets of G 1 ad G respectively if ad oly if D 1 [ D is a doiatig set of G 1 [ G : It holds that D 1 [ D = D 1 + D : The (G 1 [ G )= i (G 1 ) (G ) which proves the theore. i+= As a cosequece, we have the followig corollary. Corollary 3.. ( S i=1 G i t)= (G 1 t) (G t) ::: (G t) : Let G 1 + G be the su of graphs G 1 =(V 1 E 1 ) ad G =(V E ) deed as G 1 + G =(V 1 [ V E) where E = E 1 [ E [ffx yg : x V 1 y V g : Theore 3.3. Let G 1 = (V 1 E 1 ) ad G = (V E ) be ay graphs such that V 1 = 1 ad V = : The (G 1 + G t)= (K 1 + t) ; t 1 [ (K t) ; (G t)] ;t [ (K 1 t) ; (G 1 t)] :

6 6 JORGE LUIS AROCHA AND BERNARDO LLANO Proof. Let D be a doiatig set of G 1 + G such that D = : We dee the followig sets: S 1 = fd V 1 [ V : D is a doiatig set i G 1 ad G g S 1 = fd V 1 : D is ot a doiatig set i G 1 g ad S 1 = fd V : D is ot a doiatig set i G g : With this otatio, we have that Therefore (G 1 + G )= 1 + S 1 = 1 + ; " 1 ; S 1 ; S1 : # ; (G 1 ) ; " # ; (G ) : Multiplyig by t ; ad suig for all =1 :::, the desired result is established. This theore ca be applied to copute the doiatig polyoial of the coplete -partite graph K 1 ::: = K 1 + K + ::: + K : (K 1 ::: t) = (1 + t) ; t ; i=1 t ; i [(1 + t) i ; t i ; 1] where = P i=1 i : Cosider ow a digraph ;=(U A) where U ad A deote the set of vertices ad arcs respectively. The sets of the ex-eighbourhood ad i-eighbourhood of a vertex x are deoted by N + (x) adn ; (x) respectively ad write N + [x]adn ; [x] for the respective closed eighbourhoods. We say that D U is a doiatig set of ; if for every vertex v U ; D there exists u D such that (u v) A that is, N ; [v] \ D 6= The doiatig polyoial of a digraph ; is deed siilarly as for graphs. The properties proved before are valid i this case too. Let us call a bipartite digraph ; = (U 1 U A) oe-way if its arcs are all directed fro part U 1 to part U : The faily of doiatig sets of the oe-way bipartite digraph ; is deoted by D ; : With this deitio, D ; U 1 : Observe thatifu 1 = the (; t) = 0 (there is o doiatig set) ad if U = the (; t) = (1 + t) by covetio. Let G =(V E) be a graph. We costruct a oe-way bipartite digraph eg = (U 1 U A) fro the graph G such that U 1 ad U are disoit copies of the set V ad A = f(i i) :i V g[f(i ) ( i) :fi g Eg :

7 Lea 3.4. (G t) = MEAN VALUE 7 eg t : Proof. It is eough to show that D G = De G : The relatio D G De G is evidet. Coversely, suppose that there exists D De G such that D = D G : The for every y = D we have that N ; e G [y] \ D 6= ad there exists y = D such that N [y] \ D =. But there exists x D such that (x y) A ad by the costructio of e G the fx yg E which isacotradictio. Theore 3.5. For ay oe-way bipartite digraph ;=(U 1 U A) ad i U 1 (; t)=t (; ; i t)+ ; ; N + [i] t : Proof. The uber of doiatig sets of cardiality i ; splits ito two parts: (i) The uber of doiatig sets of cardiality ot cotaiig vertex i i.e. the uber of doiatig sets of cardiality i ; ; i =(U 1 ; i U A; A 0 ) where A 0 = f(i i)g [f(i ) : U g : (ii) The uber of doiatig sets cotaiig vertex i: Let i D where D is a doiatig set ad D = : Delete vertex i ad all the vertices doiated by it.the the uber of doiatig sets of cardiality is equal to the uber of those sets, but of cardiality ; 1 i ; ; N + [i] = (U 1 ; i U ; N + (i) A; A 00 ) where A 00 = A 0 [ (x y) :x N ; (y) y N + (i) o These sets ca be chose i ;1 (; ; N + [i]) ways. Therefore (;) = (; ; i)+ ;1 ; ; N + [i] : Multiplyig this equality by t ; ad suig for all =1 :::, we obtai the result. Observe that lea 3.4 ad theore 3.5 iply that the recurrece holds for ay graph G: (G t) =t eg ; i t + eg ; N + [i] t

8 8 JORGE LUIS AROCHA AND BERNARDO LLANO 4. Average doiatig polyoial Let us cosider the doiatig polyoial (; t) of a oe-way bipartite digraph ; as a rado variable, whose average value i the faily D of all labeled bipartite graphs with partite sets of size ad respectively, is dee by (4.1) (t) = 1 ;D (; t) : This polyoial is called the average doiatig polyoial of the faily D. Theore 4.1. (t) = ;1 = ; t, 1: Proof. Let ; =(U 1 U A) D : Applyig theore 3.5 to (4.1), we obtai that (4.) (t) = t ;D (; ; i t)+ 1 ;D ; ; N + [i] t : Observe that (; ; i t) = ;D ;D ;1 (; t) sice vertex i ca be oied to each oe-way bipartite digraph of D ;1 i ways. O the other had, if N + (i) = the the secod su i the right of (4.) rus through all labeled oe-way bipartite digraphs of the faily D ;1 ; : The labels of the vertices of N + (i) U ca be chose i ways, there are o edges betwee the ad these vertices ca be oied to the ;1 vertices of the set U 1 i (;1) ways. The ;D (; ; N [i] t) = = =0 (;1) ;D ;1 ; (;1)(;) ;D ;1 (; t) (; t) :

9 Therefore ad so (4.3) (t) = t (;1) + 1 MEAN VALUE 9 ;D ;1 (t) =t ;1 (t)+ 1 (; t) (;1)(;) ;D ;1 ;1 (t) : (; t) Let us cosider the followig expoetial geeratig fuctio: (x y t) = 0 0 (t) x y : Multiplyig (4.3) by x y = ad suig for all 0 we have that (t) x y = t ;1 (t) x y Fro this forula, (4.4) (x y 0 0 = x y = t (x y t)+e y x y t x y x ;1 (t) x = ;1 (t) 0 0 = e y x y t : y ;1 (t) y 0 y ; ( ; ) ;1 (t) : Let us d the solutio of the partial dieretial equatio (4.4) i the followig for (4.5) (x y t) =e y f (x y t) :

10 10 JORGE LUIS AROCHA AND BERNARDO LLANO Cosequetly, (x y f (x y t) = 0 = tf (x y t)+f 0 x y t : f (t) x y the f (t) = t + 1 f ;1 (t) = t + 1 ;1 Coputig f 1 (t) we have fro (4.5) that 1 (t) = f 1 (t) : f 1 (t) : Sice 1 (t) =1= for 1ad 1 0 (t) =1+t the f 1 (t) = (;1) 1 (t) =(;1) t + 1 : Therefore ad (t) = f (t) =(;1) t + 1 (;1) t + 1 ;1 = = ; t : Observe that the recurrece for the doiatig polyoial is closed i the faily D : The probles of dig a recurrece relatio for the doiatig polyoial which is closed i the faily G ad the calculatio of the average polyoial i this faily reai ope. The sae questios ca be posed for polyoials deed for other ivariats of graphs, such as iial doiatig sets, K -doiatig sets (see [7]), vertex-coverigs ad edge-coverigs. Refereces [1] C. Berge, Graphs ad Hypergraphs (North-Hollad, Lodo, 1973). [] E.J. Farrell, A itroductio to atchig polyoials, J. Cobiatorial Theory Ser B 7 (1979) [3] C.D. Godsil ad I. Guta, O the theory of the atchig polyoial, J. Graph Theory 5 (1981) [4] C.D. Godsil, Algebraic Cobiatorics (Chapa ad Hall, New Yor, 1993).

11 MEAN VALUE 11 [5] O.J. Heila ad E.H. Lieb, Mooers ad diers, Phys. Rev. Lett. 4 (1970) [6] O.J. Heila ad E.H. Lieb, Theory of ooer-dier systes, Co. Math. Phys. 5 (197) [7] M.A. Heig, O.R. Oellera ad H.C. Swart, The diversity of doiatio, Discrete Math. 161 (1996) [8] N.N. Lebedev, Special Fuctios ad their Applicatios (Dover, New Yor, 197). [9] L. Lovasz, Cobiatorial Probles ad Exercises (North-Hollad, Asterda, 1979). [10] O. Ore, Theory of Graphs (Aer. Math. Soc., Providece, 196). Istituto de Mateaticas, UNAM, Circuito Exterior, Ciudad Uiversitaria, Mexico, D.F E-ail address: arocha@ath.ua.x E-ail address: bllao@ath.ua.x