Formulas for the Number of Spanning Trees in a Maximal Planar Map

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1 Applied Mathematical Scieces Vol o Formulas for the Number of Spaig Trees i a Maximal Plaar Map A. Modabish D. Lotfi ad M. El Marraki Departmet of Computer Scieces Faculty of Scieces Uiversity of Mohamed V P.O. Box 1014 Rabat Morocco hafizmod@yahoo.fr dou.lotfi@ gmail.com marraki@fsr.ac.ma Abstract The umber of spaig trees of a map C deoted by τc is the total umber of distict spaig subgraphs of C that are trees. A maximal plaar map is a simple graph G formed by vertices 3 edges ad all faces havig degree 3 []. I this paper we derive the explicit formula for the umber of spaig trees of the maximal plaar map ad deduce a formula for the umber of spaig trees of the crystal plaar map. Mathematics Subject Classificatio: 05C85 05C30 Keywords: graphs; maps; maximal plaar map; crystal plaar map; fa plaar map; complexity; spaig trees 1 Itroductio First of all let s recall some ecessary defiitios related to our work a udirected graph G is a triplet V G E G δ where V G is the set of vertices of the graph G E G is the set of edges of the graph G ad δ is the applicatio δ : E G Pv e i δe i ={v j v k } with v j ad v k are ed vertices of the edge e i. A loop is a edge e i E G with v j = v k ifδe i =δe j with i j the the edges e i ad e j are called multiple. A graph which cotais either multiple edges or loops is called a simple graph. The degree of a vertex v oted degv is the umber of edges icidet to it. The sum of the degrees of all vertices of a graph is equal to twice the umber of its edges i.e. v V G degv = E G. A graph G is called coected if ay two of its vertices may be coected by a path [1]. The graphs that we cosider are i most cases coected but may cotai multiple edges. A map C is a graph G draw o a surface X

2 3148 A. Modabish D. Lotfi ad M. El Marraki or embedded ito it that is a compact variety -dimesioal orietable i such a way that: the vertices of graph are represeted as distict poits of the surface the edges are represeted as curves o the surface that itersect oly at the vertices if we cut the surface alog the graph thus draw what remais that is the set X\G is a disjoit uio of coected compoets called faces each homeomorphic to a ope disk [6]. A plaar map is a map draw o the plae. Through this paper all maps are plaar ad coected. Euler s formula for maps is: V C + F C E C = g where F C is the set of faces of the map C ad g is the geus of a map C i the plaar case g =0. [6]. A tree is a coected graph without cycle. A pla tree is a tree desiged i the plae. The distace betwee two distict vertices v i ad v j of a tree deoted by dv i v j is equal to the legth of the shortest path umber of edges i that coects v i ad v j by Covetioally dv i v i = 0. We defie a complete vertex i a graph G by the vertex v 0 such that du v 0 = 1 for each u V G \{v 0 }. I a complete graph all the vertices are completes. The Wieer idex of a coected graph is the sum of distaces betwee all pairs of vertices [3] [4] the Wieer idex of a coected graph G is defied as: W G = dv i v j. {v i v j } V G The particular case of trees has bee echoed by several people. For a tree with vertices the Wieer idex W T is miimized by the star tree with vertices ad maximized by the path with vertices [1]. I [3] we worked o the Wieer idex i the case of plaar maps ad gave a iequality which miimizes ad maximizes ay map by the maximal plaar map with vertices ad the path of vertices ad also we gave a formula which calculates the Wieer idex of maximal plaar map E our results give i the followig theorem: Theorem 1. Let C be a map with vertices the +=W E W C W P = 1. 6 The complexity of a map C deoted by τc is the umber of spaig trees of the map C. A spaig tree is a subgraph which is a tree that visits all vertices see Fig. 1. The umber of spaig trees i a graph etwork G is a importat ivariat of the graph etwork. It is also a importat measure of reliability of a graph etwork. A classic result of Kirchhoff [5] [9] ca be used to determie the umber of spaig trees for C =V C E C. Let V C = {v 1 v... v } to state the result we

3 Formulas for the umber of spaig trees 3149 Figure 1: A map C gives rise to 16 spaig trees defie the characteristic matrix L =[a ij ] as follows: i a ij = pv i v j if i j where pv i v j is the umber of edges that coects v i with v j ii a ij equals the degree of vertex v i if i = j ad iii a ij = 0 otherwise. The Kirchhoff matrix tree theorem states that all cofactors of L are equal ad their commo value is τc. The matrix tree theorem ca be applied to ay map C to determie τc but this requires evaluatig a determiat of a correspodig characteristic matrix. However for a few special families of graphs there exist simple formulas which make it much easier to calculate ad determie the umber of correspodig spaig trees especially whe these umbers are very large. Oe of the first such results is due to Cayley who showed that complete graph o vertices K has spaig trees [1] that is he showed τk = for. Theorem. [1] If L C is a matrix obtaied by deletig row i ad colum j of the Laplacia matrix LC the τc = 1 i+j det L C. Aother result is due to Sedlacek [10] who derived a formula for the wheel o + 1 vertices W +1 which is formed from a cycle C o vertices by addig a vertex adjacet to every vertex of C see Fig.. I particular he showed that τw +1 = for 3. I [8] we gave a formula to calculate the complexity of the -Grid chais G where is the umber of squares V G = + see Fig. is as follows: τg =

4 3150 A. Modabish D. Lotfi ad M. El Marraki also aother formula to calculate the complexity of the -Fa chais F where is the umber of triagles V F = + see Fig. is as follows: τf = Figure : The Wheel the -Grid chais ad the -Fa chais I additio other formulas are related to our work ca also be foud i [7] [10] [11]. I this paper we derive the explicit formula for the umber of spaig trees of a maximal plaar map ad deduce a formula for the umber of spaig trees of the crystal plaar map. Mai Results Before presetig the mai results we eed the followig lemmas: Lemma 1. Let C be a map ad τc deote the umber of spaig trees of C. Ife = v i v i+1 EC e is ot a loop the τc =τc e+τc.e. Proof: See [8] [1]. Let C be a map of type C= C 1 C C is a map such that v 1 ad v two vertices of C coected by a edge e []see Fig. 3. Figure 3: A map C= C 1 C

5 Formulas for the umber of spaig trees 3151 Property 1. Let C be a map of type C = C 1 C - C 1 ad C have two commo vertices v 1 v a commo edge e ad a commo face the exteral face. - V C =V C1 +V C - E C =E C1 +E C -1 ad F C =F C1 +F C -1. Example 1. Here is a example of maps C C 1 C C 1 e C e C 1.e ad C.e see Fig. 4. Figure 4: A example of maps C C 1 C C 1 e C e C 1.e ad C.e Lemma. Let C be a map of type C= C 1 C see Fig. 3 the τc =τc 1 τc τc 1 e τc e. Proof: See [8]. Defiitio 1. A maximal plaar map [] Let E be a family of maps that cotais: vertices two complete vertices of degree 1 two vertices of degree 3 ad 4 vertices of degree 4 faces of degree 3 all faces havig degree 3 3 edges the the family of this maps is called a maximal plaar map. The maps E 3 ad E 4 are preseted i the example 3. For E 5 E 6 ad E see Fig. 5 Example. I the maps E 3 ad E 4 all the vertices are complete. I other words we say that the map is complete see Fig 6.

6 315 A. Modabish D. Lotfi ad M. El Marraki Figure 5: The maps E 5 E 6 ad E Figure 6: The maps E 3 ad E 4 Lemma 3. Let E be a maximal plaar map with vertices the /τe 3. Proof: We first form the Kirchhoff characteristic matrix L correspodig to the labelig of E show as follows i Figure 7: Figure 7: The pricipal matrix of E Next we focus our attetio o the pricipal submatrix L 1 which obtaied by cacelig its last row ad colum correspodig to vertex v. So the umber of spaig trees of a maximal plaar map E equals τe =detl 1.

7 Formulas for the umber of spaig trees τe = we deote r i by the i-th row ad c i by the i-th colum of the determiat. I the previous determiat we replace c 1 by c 1 + c c 1 i.e. we add to the first colum the sum of other trasformatio is symbolized as follows: c 1 1 i=1 c i ; this does ot chage the determiat the we obtai: τe = Next we replace c j by c 1 + c j for j =... 1 i.e. c j c 1 + c j we obtai: τe = Expadig L 1 alog the first row we obtai the determiat of order ad expadig the determiat obtaied alog the first row we obtai the determiat of order 3 3 as follows:

8 3154 A. Modabish D. Lotfi ad M. El Marraki τe = hece the result. Lemma 4. Let E be a maximal plaar map with vertices the τe = 4τE 1 1 τe 5. Proof: By lemma 3 we have the determiat of order 3 3: τe = I the previous determiat we deote by c i by the i-th colum ad r i by the i-th row of the determiat c i c i c i+1 for i = we obtai the determiat as follows: τe = I the ed r i i j=1 r j for i =... 3 we obtai the determiat as follows:

9 Formulas for the umber of spaig trees 3155 τe = =4Δ 1 Δ where Δ i = detd i Δ i =4Δ i 1 Δ i because Δ i is tri-diagoal matrix ad D i is defied as followig: D i = the τe i i With the same techique we obtai: hece the result. τe 1 1 =Δ 1 ad τe =Δ 3 Applicatios Theorem 3. A maximal plaar map The complexity of the maximal plaar map E see Fig. 5 is give by the followig formula: τe = Proof: τe 3 =3τE 4 = 16 τe 5 = 75 by lemma 4 hece we obtai the system: τe = 4τE 1 1 τe τe = 4τE 1 1 τe 3 =3 τe 4 =16 τe with The we get a sequece such that u =4u 1 u 3 therefore the characteristic equatio is r 4r + 1 = 0 so the solutios of this equatio are:

10 3156 A. Modabish D. Lotfi ad M. El Marraki r 1 = 3 ad r =+ 3 hece: τe =α 3 + β + 3 αβ R 1. Usig the iitial coditios τe 3 3 = 1 ad τe 4 4 = 4 we obtai: α = β = hece the result. Let F ad G be the maps as follows see Figure 8: Figure 8: The maps F ad G Theorem 4. A Fa plaar map The complexity of the Fa plaar maps F ad G see Fig. 8 is give by the followig formulas: τf = τg = Proof: We put f = τf ad g = τg. f =g = 1 i the map F we cut the first cycle see Fig. 8 ad we use Lemma the same goes for the map G the we obtai: τg =3τF 1 τg 1 τf =τg τf 1 therefore we have the followig system: { f =g f 1 g =3f 1 g 1 with f = ad g =1

11 Formulas for the umber of spaig trees 3157 we replace by the value of g i the first equatio we get: { f =5f 1 g 1 g =3f 1 g 1 with f = ad g =1 f g f = M g = M f 1 g 1 f 1 g where M = 3-1 =... = M f g the we compute M : det M λi =λ 4λ +1=0λ 1 = 3 ad λ =+ 3 λ 1 λ the there is P ivertible such that M = PDP 1 where λ1 0 D = 0 λ P is the trasformatio matrix formed by eigevectors 1 1 P = M = PD P 1 where D = P 1 = from which we obtai M the f g = M f g hece the result. Let E be a maximal plaar map show i Fig. 5. If we delete the edge e = v 1 v of E we obtai the crystal plaar map C with vertices see Fig. 9.

12 3158 A. Modabish D. Lotfi ad M. El Marraki Figure 9: The crystal plaar map C Theorem 5. A crystal plaar map The complexity of the crystal plaar map C see Fig. 9 is give by the followig formula: τc = Proof: By Lemma 1 we have: τe =τe e+τe.e sice τe e = τc ad τe.e =τf see Fig. 10: Figure 10: The maps E E e ad E.e the τc =τe τf. We replace by the value of τe from Theorem 3 ad the value of τf 1 from Theorem 4 thus our result follows. 4 Coclusio I this paper we have iterested by calculatig the umber of spaig trees of some particular plaar maps for that we have derived the explicit formula to calculate the umber of spaig trees of the maximal plaar map the we have deduced the formula for the umber of spaig trees of the crystal plaar map.

13 Formulas for the umber of spaig trees 3159 Refereces [1] G. A. Cayley A Theorem o Trees. Quart. J. Math [] R. Diestel Graph Theory. Spriger-Verlag New York 000. [3] M. El Marraki A. Modabish Wieer idex of plaar maps. Joural of Theoretical ad Applied Iformatio Techology JATIT Vol o [4] M. El Marraki G. Al Hagri Calculatio of the Wieer idex for some particular trees. Joural of Theoretical ad Applied Iformatio Techology JATIT Vol. 010 o [5] G. G. Kirchhoff Über die Auflösug der Gleichuge auf welche ma bei der Utersuchug der lieare Verteilug galvaischer Strme gefhrt wird A. Phys. Chem [6] S.K. Lado A. Zvoki Graphs o Surfaces ad Their Applicatios. Spriger-Verlag 004. [7] A. Modabish D. Lotfi ad M. El Marraki The umber of spaig trees of plaar maps: theory ad applicatios. Proceedigs of the Iteratioal Coferece o Multimedia Computig ad Systems IEEE 011 Ouarzazate-Morocco to appear. [8] A. Modabish M. El Marraki The umber of spaig trees of certai families of plaar maps. Applied Mathematical Scieces Vol o [9] R. Merris Laplacia matrices of graphs A survey Li. Algebra Appl [10] J. Sedlacek Lucas Numbers i Graph Theory. I: Mathematics Geometry ad Graph Theory Chech. Uiv. Karlova Prague [11] J. Sedlacek O the Spaig Trees of Fiite Graphs Cas. Pestovai Mat [1] D. B. West Itroductio to Graph Theory. Secod Editio Pretice Hall 00. Received: April 011