A Witt type formula. G.A.T.F.da Costa 1 Departamento de Matemática Universidade Federal de Santa Catarina Florianópolis-SC-Brasil.

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1 A Wtt type formula G.A.T.F.da Costa 1 Departamento de Matemátca Unversdade Federal de Santa Catarna Floranópols-SC-Brasl arxv: v2 [math.co] 4 Mar 2013 Abstract Gven a fnte, connected graph G, T ts edge adjacency matrx, and, a postve nteger, ths paper nvestgates some combnatoral and algebrac propertes of the Wtt type formula Ω(, T ) = 1 µ(g) T rt g. The sum ranges over the postve dvsors of and µ s the Möbus functon. 1 Introducton The objectve of the present paper s to nvestgate algebrac and combnatoral aspects of the formula gven n the abstract whch gves the number of equvalence classes of arbtrarly orented but non-bactracng non-perodc closed paths of length n a orented graph. The formula s well nown n assocaton wth the zeta functon of a graph nvestgated by several authors. See [8, 9, 10] and references theren. As shown n ths paper the formula has several propertes of the Wtt formula type not nvestgated prevously, as far as I now. Let s recall Wtt formula and some of ts propertes. Let be a postve nteger, R a real number, µ the classcal Möbus functon defned by the rules: a) µ(+1) = +1, b) µ(g) = 0, f g = p e p eq q, p 1,..., p q prmes, and any e > 1, c) µ(p 1...p q ) = ( 1) q. The polynomal of degree n R wth ratonal coeffcents gven n terms of Möbus functon, M(; R) = 1 µ(g)r g, (1.1) has many applcatons n algebra and combnatorcs [5]. It s called the Wtt formula when t s assocated wth the followng result [6]: If V s an R-dmensonal vector space and L s the free Le algebra generated by V then L has a Z >0 gradaton 1 g.costa@ufsc.br 1

2 L = L, L has dmenson gven by M(; R) whch satsfes the formal relaton (1 z ) M(;R) = 1 Rz (1.2) called the Wtt dentty. otce that the coeffcent of the lnear term n the rght hand sde s mnus the dmenson of the vector space that generates the Le algebra. Wtt dentty follows from the Poncaré-Broff-Wtt theorem whch says that (1 z ) M(;R) = 1 + R z (1.3) s the generatng functon for the dmensons of the homogeneous subspaces of the envelopng algebra of L. Wtt formula s also called the neclace polynomal because M(; R) gves the number of nequvalent non-perodc colorngs of a crcular strng of beads - a neclace - wth at most R colors [12]. In [7] S. Sherman assocated t to the number of equvalence classes of closed non-perodc paths of length whch traverse counterclocwsely wthout bactracng the edges of a graph wth R loops counterclocwsely orented and hooed to a sngle vertex so that Ω generalzes M. otce that the coeffcent of the lnear term n the rght hand sde of Wtt dentty s mnus the number of loops n the graph. It s proved that the formula for Ω(, T ) satsfes some denttes analogous to those satsfed by M(, R) whch Carltz proved n [11] and Metropols and Rota n [12]. In [5], Moree proved smlar denttes for hs Wtt transform. Also, the formula can be nterpreted as a dmenson formula and t can be assocated to a colorng problem. The paper s organzed as follows. In secton 2, some prelmnary defntons and results are gven. In secton 3, several denttes satsfed by the formula s proved. In secton 4, the formula s nterpreted as a dmenson formula assocated to free Le super algebras and, n secton 5, to neclace colorngs. The formula s appled to some examples. 2 Prelmnares Let G = (V, E) be a fnte connected and orented graph where V s the set of vertces wth V elements and E s the set of orented edges wth E elements labeled e 1,...,e E. An edge has an orgn and an end as gven by ts orentaton. The graph may have multple edges and loops. ow, consder the graph G bult from G by addng n the opposng orented edges e E +1 = (e 1 ) 1,...,e 2 E = (e E ) 1, (e ) 1 beng the orented edge opposte to 2

3 e and wth orgn (end) the end (orgn) of e. In the case that e s an orented loop, e + E = (e ) 1 s just an addtonal orented loop hooed to the same vertex. Thus, G has 2 E orented edges. A path n G s gven by an ordered sequence (e 1,..., e ), {1,..., 2 E }, of orented edges n G such that the end of e s the orgn of e +1. Also, a path can be represented by a word n the alphabet of the symbols n the set {e 1,..., e 2E }, a word beng a concatenated product of symbols whch respect the order of the symbols n the sequence. In ths paper, all paths are cycles. These are non-bactracng tal-less closed paths, that s, the end of e concdes wth the orgn of e 1, subjected to the nonbactracng condton that e +1 e + E. In another words, a cycle never goes mmedately bacwards over a prevous edge. Tal-less means that e 1 e 1. The length of a cycle s the number of edges n ts sequence. A cycle p s called perodc f p = q r for some r > 1 and q s a non perodc cycle. umber r s called the perod of p. The cycle (e, e 1,..., e 1 ) s called a crcular permutaton of (e 1,..., e ) and (e 1,..., e 1 1 ) s an nverson of the latter. A cycle and ts nverse are taen as dstnct. The classcal Möbus nverson formula s used several tmes n ths paper. Gven arthmetc functons f and g t states that g(n) = d n f(d) f and only f f(n) = d n µ(d)g(n/d). In order to count cycles of a gven length n a graph G a crucal tool s the edge adjacency matrx of G [8]. Ths s the 2 E 2 E matrx T defned as follows: T j = 1, f end vertex of edge s the start vertex of edge j and edge j s not the nverse edge of ; T j = 0, otherwse. Theorem 2.1. ([8]) The number T rt (over)counts cycles of length n a graph G. Proof. Let a and b be two edges of G. The (a, b) th entry of matrx T s (T ) (a,b) = e 1,...,e 1 T (a,e1 )T (e1,e 2 )...T (e 1,b) From the defnton of the entres of T t follows that (T ) (a,b) counts the number of paths of length wth no bactracs from edge a to edge b. For b = a, only closed paths are counted. Tang the trace gves the number of non-bactracng closed paths wth every edge taen nto account as startng edge, hence, the trace overcounts closed paths because every edge n the path s taen nto account as startng edge. The paths counted by the trace are tal-less, that s, e 1 e 1 ; otherwse, T rt = a (T ) (a,a) would have a term wth entry (a, a 1 ) whch s not possble. 3

4 Theorem 2.2. [9] Denote by Ω(, T ) the number of equvalence classes of non perodc cycles of length n G. Ths number s gven by the followng formula: Ω(, T ) = 1 µ(g) T rt g (2.1) Proof. In the set of T rt cycles there s the subset wth Ω(, T ) elements formed by the non perodc cycles of length plus ther crcular permutatons and the subset wth g 1 Ω(, T ) elements formed by the perodc cycles of length (whose g g perods are the common dvsors of ) plus ther crcular permutatons. (A cycle of perod g and length s of the form (e 1 e 2...e α ) g where α = /g, and (e 1 e 2...e α ) s a non perodc cycle so that the number of perodc cycles wth perod g plus ther crcular permutatons s gven by (/g)ω(/g, T )). Hence, T rt = ( ) g Ω g, T Möbus nverson formula gves the result. Remar 1. Some terms n the rght hand sde of (2.1) are negatve. In spte of that the rght hand sde s always postve. Multply both sdes by. The fst term equals T rt whle the other terms gve (n absolute value) the number accordng to perod of the varous subsets of perodc cycles whch are proper subsets of the larger set wth T rt elements. Remar 2. Wtt formula can be expressed n a form analogous to (2.1). Defne Q as the R R matrx wth all entres equal to one. The trace T rq = R counts counterclocwsely orented cycles n the graph wth R loops counterclocwsely orented and hooed to a sngle vertex so that M(; R) = 1 µ(g)t rq g (2.2) A recurrence relaton whch may be useful n practcal calculatons of Ω(, T ) s gven next. Theorem 2.3. Ω(, T ) = T rt,g g Ω (g, T ) (2.3) Proof. Ths follows from T rt = g Ω (g, T ) = Ω(, T ) + g Ω (g, T ).,g 4

5 3 Some denttes satsfed by Ω It turns out that Ω(, T ) satsfes some denttes analogous to those satsfed by Wtt formula proved n [11, 12]. These denttes are establshed n ths secton. In [5], Moree proved smlar denttes for hs Wtt transform. Theorem 3.1. Gven the matrces T 1 and T 2 defne S(s, T ) = g s µ(g) T rt g, = 1, 2, and denote by T 1 T 2 the Kronecer product of T 1 and T 2. Then, S(s, T 1 )S(t, T 2 ) = S(, T 1 T 2 ) (3.1) [s,t]= The summaton s over the set of postve ntegers {s, t [s, t] = }, [s, t] beng the least common multple of s, t. It also holds that S(, T l ) = S(t, T ) (3.2) [l,t]=l Proof. In order to prove (3.1) t suffces to consder the equvalent formula (see [11]) S(, T 1 T 2 ) [s,t]= S(s, T 1 )S(t, T 2 ) = Usng Möbus nverson formula, the left hand sde s equal to S(s, T 1 ) S(t, T 2 ) = (T rt1 )(T rt2 ) s t But (T rt1 )(T rt2 ) = T r(t 1 T 2 ). By Möbus nverson formula ths gves the rght hand sde of the equvalent formula. Usng deas from [5], the next dentty can be proved usng the followng equvalent formula: S(t, T ) = S(, T l ) [l,t]= l g The left hand sde s equal to l t l S(t, T ) = T rt = T r(t l ). Apply Möbus nverson formula to get the result. Remar 3. Formula (3.1) may be generalzed to the case T = T 1 T 2... T l to gve S(s 1, T 1 )... S(s l, T l ) = S(, T ) (3.3) [s 1,...,s l ]= Also, t can be proved that S(, T s 1 T r 2 ) = [rp,sq]=nrs 5 S(p, T 1 )S(q, T 2 ) (3.4)

6 where r and s are relatvely prme and the summaton s over all postve ntegers p, q such that [rp, sq] = nrs. The proof s an applcaton of prevous denttes as n [12], Theorem 5. Remar 4. In terms of Ω, usng that [s, t](s, t) = st, (7) becomes (s, t)ω(s, T 1 )Ω(t, T 2 ) = Ω(, T 1 T 2 ) (3.5) [s,t]= where (s, t) s the maxmum common dvsor of s and t. Ths can be extended to the general case (3.3) to gve (s 1,, s l )Ω(s 1, T 1 )... Ω(s l, T l ) = Ω(, T ) (3.6) [s 1,...,s l ]= where (s 1,, s l ) s the greatest common dvsor of (s 1,, s l ) and the sum runs over all postve ntegers (s 1,, s l ) wth least common multple equal to and T = T 1 T l. Also, from (8), In terms of Ω (10) reads Ω(, T l ) = Ω(, T s 1 T r 2 ) = [l,t]=l [rp,sq]=rs t Ω(t, T ). (3.7) pqω(p, T 1 )Ω(q, T 2 ) Usng (rp, sq)[rp, sq] = rpsq wth [rp, sq] = rs mples (rp, sq) = pq and Ω(, T1 s T2 r ) = (rp, sq)ω(p, T 1 )Ω(q, T 2 ) [rp,sq]=rs Replace s and r by s/(r, s) and r/(r, s) to get (r, s)ω(, T s/(r,s) 1 T r/(r,s) 2 ) = (rp, sq)ω(p, T 1 )Ω(q, T 2 ) (3.8) The sum s over p, q such that pq/(pr, qs) = /(r, s). Another dentty satsfed by Ω s of the Wtt type (1.2). It s a well nown result about the ζ functon of a graph G whch s defned as follows: ζ(z) := [p] (1 z l(p) ) 1 = (1 z ) Ω(,T ) (3.9) 6

7 See [8] and [9]. The frst product s over the equvalence classes [p] of bactracless and tal-less closed paths of length l(p) n G. It s a famous result that ζ = [det(1 zt )] 1, hence, Ω(, T ) satsfes the Wtt type dentty (1 z ) Ω(,T ) = det(1 zt ) (3.10) As mentoned n the ntroducton, the coeffcent of the lnear term n Wtt s dentty s the negatve of the number of loops n a graph wth R loops hooed to a sngle vertex. The coeffcents n the expanson of the determnant det(1 zt ) as a polynomal n z also have nce combnatoral meanngs related to the structure of the graph G as proved by several authors. See [10] and references theren. ext theorem gves formulas for these coeffcents and for those of the nverse of the determnant whch are relevant for the next secton. Theorem 3.2. Defne Then, T rt g(z) := z. (3.11) where + (1 z ) ±Ω(,T ) = e g(z) = [det(1 zt )] ± = 1 c ± () = λ ± (m) m=1 a 1 + 2a a = a a = m =1 + =1 c ± ()z, (3.12) (T rt ) a a! a (3.13) wth λ + (m) = ( 1) m+1, λ (m) = +1, c + () = 0 for > 2 E, and c () 0. Furthemore, T rt = s = (s ) 1, s Z 0 s = s +1 ( s 1)! (±1) c± () s (3.14) s! where s = s, s! = s!. Proof. Defne P ± by P ± (z) = + =1 (1 z ) ±Ω(,T ) 7

8 Tae the logarthm of both sdes and use (2.1) to get lnp ± = = + T rt 1 + Ω(, T )z = z = g(z) ( ) 1 Ω, T z from whch the frst equalty n (3.12) follows. From the defnton of g(z), t follows that T rt + 1 g(z) := z = T r T z = ±T r ln(1 zt ) = ±ln det(1 zt ) provng the second equalty n (3.12). The thrd equalty s obtaned formally expandng the exponental. formal Taylor expanson of 1 e g, the coeffcents c ± are gven by d As the c ± () = 1 [ ±(1 e g ) ]! dz z=0 Usng Faa d Bruno s formula as n [1], the dervatves can be computed explctly and (3.13) follows. The determnant s a polynomal of maxmum degree 2 E, hence, c + () = 0 for > 2 E. Clearly, c () 0. To prove (3.14) wrte [2] ( ln 1 ) c ± ()z = = ± + l=1 + l=1 ( 1) l (±1) l ( ± ) l c ± ()z ( s )! (s!) s = (s ) 1 s Z 0 s = l ( c± () s ) z s l = + z =1 s = (s ) 1, s Z 0 s = s +1 ( s 1)! (±1) c± () s s! The second equalty n (3.12) appled to the left hand sde yelds ( ln 1 ) + c ± ()z T rt = z 8 =1

9 Compare coeffcents to get the result. Remar 5. Wtt dentty can be expressed n terms of a determnant: (1 z ) M(;R) = 1 Rz = det(1 zq) The proof s analogous to the proof of prevous theorem usng (2.2). 4 Ω and free Le super algebras As mentoned n the ntroducton the coeffcent n the Wtt formula (1.1) has an algebrac nterpretaton as the negatve of the dmenson of a vector space that generates a free Le algebra and the nverse of ths formula s the generatng functon of dmensons of the subspaces of the envelopng algebra of the Le algebra. Is t possble that the coeffcents n the determnant det(1 zt G ) above have smlar nterpretaton? The answer s postve. Formula (2.1) and the coeffcents of the determnant and ts nverse can be nterpreted algebracally n terms of data related to Le super algebras. In seres of papers S. -J. Kang and M. -H Km [2, 4] generalzed Wtt formula (1.1) to the case the free Le algebra L s generated by an nfnte graded vector space. They obtaned a generalzed Wtt formula for the dmensons of the homogeneous subspaces of L whch satsfes a generalzed Wtt dentty. In [4] S. -J. Kang extended these results to super spaces and Le super algebras. Some of the results are summarzed n the followng proposton. Proposton 4.1. Let V = =1 V be a Z >0 -graded super space wth fnte dmensons dmv = t() and super dmensons DmV = t() Z, 1. Let L = L be the free Le super algebra generated by V wth a Z >0 -gradaton nduced by that of V. Then, the super dmensons of the subspaces L are gven by formula DmL = µ(g) g W ( ) g The summaton ranges over all common dvsors g of and W s gven by W () = s T () ( s 1)! s! (4.1) t() s (4.2) where T () = {s = (s ) 1 s Z 0, s = } and s = s, s! = s!. The numbers DmL satsfy the dentty (1 z ) DmL = 1 t()z. (4.3) 9 =1

10 The rght hand sde of (4.3) s related to the generatng functon for the W s, by the relaton Furthermore, g(z) := W (n)z n (4.4) n=1 e g(z) = 1 t()z (4.5) =1 ( ) DmL 1 = z DmU(L) z (4.6) where DmU(L) s the dmenson of the -th homogeneous subspace of U(L), the unversal envelopng algebra of L. In [4], (4.1) s called the generalzed Wtt formula; W s called Wtt partton functon; and (4.3) s called generalzed Wtt dentty. See secton 2.3 of [4]. Gven a formal power seres + =1 t z wth t Z, for all 1, the coeffcents n the seres can be nterpreted as the super dmensons of a Z >0 -graded super space V = =1 V wth dmensons dmv = t and super dmensons DmV = t Z. Let L be the free Le super algebra generated by V. Then, t has a gradaton nduced by V and the homogeneous subspaces have dmenson gven by (4.1). Apply ths nterpretaton to the determnant det(1 zt ) whch s a polynomal of degree 2 E n the formal varable z. It can be taen as a power seres wth coeffcents t = 0, for > 2 E. Comparson of the formulas n Theorem 3.2 wth formulas n the above Proposton mples the next result: Proposton 4.2. Gven a graph G, T ts edge matrx, let V = 2 E =1 V be a Z >0 - graded super space wth fnte dmensons dmv = c + () and the super dmensons DmV = c + () gven by (3.13), the coeffcents of det(1 zt ). Let L = L be the free Le super algebra generated by V. Then, the super dmensons of the subspaces L are gven by DmL = Ω(, T ). The algebra has generalzed Wtt dentty gven by (3.10). The zeta functon of G (3.9) s the generatng functon for the dmensons of the subspaces of the envelopng algebra U(L) of L whch are gven by DmU(L) n = c (n), c (n) gven by (3.13). Example 1. G 1, the graph wth R 2 edges counterclocwsely orented and hooed to a sngle vertex shown n Fgure 1. The edge matrx for G 1 s the 2R 2R symmetrc matrx ( ) A B T G1 = B A =1 10

11 Fgure 1: Graph G 1 where A s the R R matrx wth all entres equal to 1 and B s the R R matrx wth the man dagonal entres equal to 0 and all the other entres equal to 1. Ths matrx has the trace gven by and the determnant T rt G 1 = 1 + (R 1)(1 + ( 1) ) + (2R 1), = 1, 2,... det(1 zt G1 ) = (1 z) [1 (2R 1)z] (1 z 2 ) R 1 = 1 2R =1 c()z where c(2r) = ( 1) R (2R 1), c(2) = ( 1) (2 1) ( R ), = 1,, R 1 and Futhermore, c(2 + 1) = 2R( 1) ( R 1 ), = 0, 1, R 1 [det(1 zt G1 )] 1 = + z q q=0 q a (2R 1) q =0 where a = =0 ( 1) ( + R 1 R 1 ) ( + R 2 R 2 ) Let s consder the case R = 2. In ths case, T rt G 1 = 2 + ( 1) + 3, det(1 zt G1 ) = 1 4z + 2z 2 + 4z 3 3z 4 11

12 so that the number of classes of reduced nonperodc cycles of length s gven by the formula Ω(, T G1 ) = 1 ) µ(g) (2 + ( 1) g + 3 g The graph generates the followng algebra. Let V = 4 =1 V be a Z >0 -graded super space wth dmensons dmv 1 = 4, dmv 2 = 2, dmv 3 = 4, dmv 4 = 3 and super dmensons DmV 1 = 4, DmV 2 = 2, DmV 3 = 4, DmV 4 = 3. Let L = L be the free graded Le super algebra generated by V. The dmensons of the subspaces L are gven by the generalzed Wtt formula DmL = 1 ) µ(g) (2 + ( 1) g + 3 g whch satsfes the generalzed Wtt dentty + (1 z ) Ω(,T G 1 ) = 1 [4z 2z 2 4z 3 + 3z 4 ] The subspace U n (L) of the envelopng algebra U(L) have dmensons gven by the zeta functon of the graph, + (1 z ) Ω(,T G 1 ) = (( 1) n n 24 12n)z n 16 n=1 so that DmU n (L) = 1 16 (( 1)n n 24 12n) Example 2. G 2, the bpartte graph shown n Fgure 2. Fgure 2: Graph G 2 12

13 The edge matrx of G 2 s T G2 = The matrx has the trace T rtg 2 = 0 f s odd and T rtg 2 = f s even, and the determnant det(1 zt G2 ) = 1 6z 2 + 9z 4 4z 6 If s odd, the number of classes of nonperodc cycles of length s Ω(, T G2 ) = 0, f s odd, and Ω(, T G2 ) = 1 µ(g)t rt g G 2 f s even. The graph generates the followng algebra. Let V = 3 =1 V 2 be a Z >0 -graded superspace wth dmensons dmv 2 = 6, dmv 4 = 9, dmv 6 = 4 and superdmensons DmV 2 = 6, DmV 4 = 9, DmV 6 = 4. Let L = L be the free graded Le superalgebra generated by V. The dmensons of the subspaces L are DmL = 0, for odd and DmL = 1 µ(g)t rt g G 2 for even. The dmensons satsfy the generalzed Wtt dentty + (1 z ) Ω(,T G 2 ) = 1 [6z 2 9z 4 + 4z 6 ] The generatng functon for the dmensons of the subspaces U n (L) of the envelopng algebra U(L) s gven by + (1 z ) Ω(,T G 2 ) = (2 2n+5 6n 14)z 2n n=1 so that DmU n (L) = 2 2n+5 6n 14 13

14 5 eclace colorngs nduced by paths Gven the set of 2 E colors {c 1,..., c 2 E }, assgn the colors c, c E + to edges e, e E + = e 1 G, respectvely, so that to a cycle of length n G corresponds an ordered sequence of colors. ow, assgn each color n ths sequence to a bead n a crcular strng wth beads - a neclace - n such a manner that two adjacent colors n the sequence are assgned to adjacent beads. The non bactracng condton for cycles mples that no two adjacent beads are panted wth colors, say, c and c E +. It s clear that there s a correspondence between the classes of nonperodc cycles of length n G and classes of nonperodc colorngs of a neclace wth beads wth at most 2 E dstnct colors nduced by the cycles so that the number of nequvalent colorngs s Ω G (, T ). Of course the structure of the graph reflects tself on the colorng. For nstance, the presence of loops n the graph means that ther assgned colors may appear repeated n a strng of adjacent beads. Ths can not happen to a color assgned to an edge whch s not a loop. The edge matrx T may be called the color matrx. It bascally tells what colors are allowed to follow a gven color n the neclace. Element T j = 1, f a color c j can follow color c and c j c E + ; T j = 0, otherwse. Example 3. The number of nonperodc colorngs of a neclace wth beads wth at most 6 colors and color matrx gven by graph G 1 wth R = 3: T G1 = s Ω(, T G1 ) = 1 ) µ(g) (3 + 2( 1) g + 5 g For = 3, Ω(3, T G1 ) = 40. The classes of nonperodc colorngs are [c c 2 j], [c 3+ c 2 j], [c c 2 3+j], [c 3+ c 2 3+j ], [c 2 c j ], [c 2 3+ c j ], [c 2 c 3+j ], [c 2 3+ c 3+j ], for (, j) = (1, 2), (1, 3), (2, 3) and [c c j c ], [c 3+ c j c ], [c c 3+j c ],[c c j c 3+ ], [c 3+ c 3+j c ], [c 3+ c j c 3+ ], [c c 3+j c 3+ ], [c 3+ c 3+j c 3+ ], for (, j, ) = (1, 2, 3), (1, 3, 2). These corresponds to the classes of cycles [e +1 e +2 j ], [e 1 e 2 j], [e +1 e 2 j ], [e 1 for (, j) = (1, 2), (1, 3), (2, 3) and [e +1 e +1 [e 1 e 1 j e +1 ], [e 1 e +1 j e 1 ],[e+1 e 1 j e 2 j ], [e +2 e +1 j ], [e 2 e +1 j ], [e +2 j e +1 ], [e 1 e +1 j e +1 ], [e+1 e 1 j e +1 e 1 ], [e 1 e 1 j e 1 e 1 j ], [e 2 e 1 j ], ],[e+1 e +1 j e 1 ], ], for (, j, ) = (1, 2, 3), (1, 3, 2). Example 4. For the graph G 2 assgn to the three orented edges e 1, e 2, e 3 the colors c 1, c 2, c 3, respectvely. The graph s bpartte so only paths of even length are possble. The number of nequvalent nonperodc colorngs nduced by the nonperodc 14

15 cycles of length s gven by Ω(, T G2 ) = 1 µ(g)t rt g G 2 (5.1) For = 2, Ω(2, T G2 ) = 6. The classes are [e 1 e 1 2 ], [e 1 1 e 2 ], [e 1 e 1 3 ], [e 1 1 e 3 ], [e 2 e 1 3 ], [e 1 2 e 3 ]. The nduced colorngs are [c 1 c 5 ], [c 4, c 2 ], [c 1 c 6 ], [c 4 c 3 ], [c 2 c 6 ], [c 5 c 3 ]. For The paths are [e 1 e 1 2 e 3 e 1 2 ], [e 1 e 1 2 e 1 e 1 3 ], [e 1 e 1 3 e 2 e 1 3 ], 2 e 3 ]. To these classes correspond the colorngs [c 1 c 5 c 3 c 5 ], [c 1 c 5 c 1 c 6 ], [c 1 c 6 c 2 c 6 ], [c 4 c 2 c 4 c 3 ], [c 4 c 2 c 6 c 2 ], [c 4 c 3 c 5 c 3 ]. The graph has no loops so strngs of two or more beads wth a same color s not possble. = 4, Ω(4, T G2 ) = 6. [e 1 1 e 2 e 1 1 e 3 ], [e 1 1 e 2 e 1 3 e 2 ], [e 1 1 e 3 e 1 Acnowledgements. I would le to than Prof. C. Storm (Adelph Unversty, USA) for sendng me hs jont paper wth G. Scott on the coeffcents of Ihara zeta functon. Also, many thans to Prof. Asterode Santana for hs help wth the fgures, latex commands and determnants. References [1] G. A. T. F. da Costa, J. Varane, Feynman dentty: a specal case revsted, Letters n Math. Phys. 73 (2005), [2] S.-J. Kang, M.-H. Km, Free Le algebras, generalzed Wtt formula, and the denomnator dentty, J. Algebra, 183 (1996) [3] S.-J. Kang, M.-H. Km, Dmenson Formula for Graded Le Algebras and ts Applcatons, Trans. Amer. Math. Soc. 351 (1999), [4] S.-J. Kang, Graded Le Superalgebras and the Superdmenson Formula, J. Algebra, 204 (1998) [5] P. Moree, The formal seres Wtt transform, Dscrete Math. 295 (2005), [6] J. P. Serre, Le Algebras and Le Groups, Benjamn, ew Yor, [7] S. Sherman, Combnatoral aspects of the Isng model for ferromagnetsm.ii. An analogue to the Wtt dentty, Bull. Am. Math. Soc. 68 (1962), [8] H. M. Star, A. A. Terras, Zeta Functons of Fnte Graphs and Coverngs, Adv. Math. 121, (1996). [9] A. A. Terras, What are prmes n graphs and how many of them have a gven length?, U.C.S.D. Math. Club, Oct. 30, Avalable at aterras/what are prmes n graphs and how many.pdf 15

16 [10] G. Scott, C. Storm, The coeffcents of Ihara zeta functon, Involve (2008), [11] L. Carltz, An arthmetc functon, Bull. Amer. Math. Soc. 43(1937) [12]. Metropols, Gan-Carlo Rota, Wtt vectors and the algebra of neclaces, Adv. Math. 50 (1983)