Graphs on Surfaces and the Partition Function of String Theory
aa r X i v : . [ m a t h - ph ] A p r Graphs on Surfaces and the PartitionFunction of String Theory
J. Manuel Garc´ıa-Islas ∗ October 24, 2018
Abstract . Graphs on surfaces is an active topic of pure mathematics be-longing to graph theory. It has also been applied to physics and relates discreteand continuous mathematics. In this paper we present a formal mathematicaldescription of the relation between graph theory and the mathematical physicsof discrete string theory. In this description we present problems of the combi-natorial world of real importance for graph theorists.The mathematical details of the paper are as follows: There is a combi-natorial description of the partition function of bosonic string theory. In thiscombinatorial description the string world sheet is thought as simplicial andit is considered as a combinatorial graph. It can also be said that we haveembeddings of graphs in closed surfaces.The discrete partition function which results from this procedure gives a sumover triangulations of closed surfaces. This is known as the vacuum partitionfunction.The precise calculation of the partition function depends on combinatorialcalculations involving counting all non-isomorphic triangulations and all span-ning trees of a graph. The exact computation of the partition function turns outto be very complicated, however we show the exact expressions for its computa-tion for the case of any closed orientable surface. We present a clear computationfor the sphere and the way it is done for the torus, and for the non-orientablecase of the projective plane.
String theory is considered to be a quantum theory of all forces of nature includ-ing of course quantum gravity [12] [18]. When strings propagate over space-timethey sweep a two dimensional surface known as world sheet. The bosonic ac-tion of this wold sheet is proportional to its area, and when propagating in ∗ e-mail: [email protected] S P = − πα ′ Z dτ dσ √− hh αβ ∂ α X µ ∂ β X ν η µν (1)which is known as the Polyakov action. The partition function of this theoryfor a fixed surface Σ is given by Z (Σ) = Z DXDg e − S P (2)where the integral is over all embeddings of the surface Σ in space-time(forexample R n ), and over all metrics on the surface.The combinatorial description of the string partition function is as follows:Consider the world sheet(two dimensional surface) Σ, and triangulate T it.Denote the vertices of the triangulated surface by latin indices i, j .Following [4], the Polyakov action can be written in a discrete form as S (Σ T )( X ) = 12 X i ∼ j ( X i − X j ) + µF ( T ) (3)where X i , X j denotes the image of vertices i, j under the embedding X of thetriangulated surface in space-time. i ∼ j means the vertices i and j are joined by an edge, µ is a parameter and F ( T ) denotes the number of triangles of the triangulation T . As explained in[4], the analogous of all metrics in the world-sheet corresponds in the discretetheory to all non-isomorphic triangulations. Therefore the partition functionfor a fixed topology Σ is given by a sum over triangulations Z (Σ) = X T Z Y i ∈ V ( T ) dX i e − S (Σ T )( X ) (4)In [2], [3] this partition function was studied. In this paper we now give a precisemathematical description which stresses the mathematical side related to graphtheory, and more specifically to problems that graph theorist are interested inat present and which are important for a deep understanding of the partitionfunction.For example, in the above partition function sum it is evident that we haveto know how to generate all non-isomorphic triangulations of an arbitrary twodimensional surface which is clearly understood in graph theory. The proce-dure is to start with the irreducible triangulations of a surface which numbergrows as the genus of the surface grows. Then an arbitrary triangulation isalways obtained from the set of irreducible ones by certain moves known asvertex splittings. The problem is that we do not know how many irreducibletriangulations there are for any surface. The most studied cases have been untilnow, the sphere, the torus, the torus of genus 2, the projective plane, the Klein2ottle which are only a few cases. It is certain that the problem becomes moredifficult as the genus of the surface grows. Recent studies on this direction havebeen considered in [1]. There has been also studies of similar problem of findingembeddings of complete graphs on surfaces [5], [11], [14].Let us mention a well known interesting combinatorial problem. When wehave a fixed triangulation of a world-sheet surface Σ, the integral Z (Σ T ) = Z Y i ∈ V ( T ) dX i e − S (Σ T )( X ) (5)is related to the well known Matrix-Tree theorem of combinatorics [10].When summing over different triangulations for a fixed closed surface Σ, wewill show how each summand of the partition function is calculated and see thatthe number of spanning trees of the triangulation is relevant. The matrix treetheorem tells us how to calculate this number for any graph. The problem isthat if the graph has numerous vertices it is not very practical to use the matrixtree theorem but an estimate number of the number of spanning trees is neededfor our calculations.Besides for a fixed number of vertices there are many non-isomorphic numberof triangulations of a surface.We divide this paper as follows. In section 2 we introduce the discrete parti-tion function of string theory and show how the partition function is calculatedfor any closed surface. We will see what each term of the sum is, even thoughthis is not sufficient to know what the sum converges to. More will be needed forthat as we will understand in this paper. In section 3 we introduce the mathe-matical concept of graphs on surfaces and the way to generate all triangulationsof a surface from the irreducible set of triangulations as well as the number ofnon-isomorphic triangulations with a fixed number of vertices and the numberof spanning trees. In section 4 we go to the main part of the paper which isto do explicit calculations on surfaces of the partition function which was ourmotivation. We consider the cases of the sphere which is the only one we cando more formally; we also show how it is done(less formally but still with rigor)for the torus and the projective plane; and finally in general the calculation forany surface. In this section we describe the discrete partition function. The nice thing aboutit is that it is completely combinatorial. Consider first a closed vacuum stringworld-sheet embedded in a space-time of dimension D . The sheet is a com-pact, connected two dimensional surface without boundaries. Let T be a non-3egenerate triangulation of it. . This means that T itself can be seen as a graph,i.e a finite collection of vertices and edges with the following properties: for anytwo different vertices it can exist one edge only which joins them; otherwisethere is no edge between two different vertices. Moreover, a single vertex cannot be joint to itself, i.e there are no loops. With these conditions we think ofthe non-degenerate triangulation of the world-sheet surface as a graph. Con-sider the discrete Polyakov action for a particular surface Σ and triangulation T which is given by equation (3).Define the combinatorial Laplacian of a graph(which extends to our trian-gulation) as follows ∆ = d if i = j − i ∼ j d is the number of edges incident to a vertex which is known as itsvalance. With this combinatorial Laplacian it is not difficult to see that thediscrete Polyakov action can be written as S (Σ T )( X ) = 12 X i ∼ j X i ∆ X j + µF ( T ) (6)In this discrete theory as mentioned in the introduction, for a fixed surface, allnon-isomorphic triangulations play the role of the metrics, and the maps whichare defined on vertices, are just the different embeddings of the triangulatedsheet in space-time. The partition function for a closed surface Σ is given by Z (Σ) = X T Z Y i ∈ V ( T ) dX i e − S (Σ T )( X ) (7)where we sum over all non-isomorphic triangulations of the surface. The mostgeneral partition function is given by summing over different topologies Z = X Σ X T Z Y i ∈ V ( T ) dX i e − S (Σ T )( X ) (8)Consider the integral (5) for a fixed triangulation T of the surface Σ. Let v be any vertex of this triangulation T and consider the graph T − v which is givenby considering the complement of the vertex v and of all the edges incident toit. Let the image of vertex v , X v be fixed. The partition function reducesthen to an integral of over all embeddings of the remaining vertices, with the We give a formal mathematical description of graphs on surfaces and in particular oftriangulations in section 4 X v is fixed. Integral (5) up to a factor can berewritten as Z (Σ T ) = e − µF ( T ) Z Y i ∈ V ( T ) dX i e P i ∼ j X i ∆ ( T − v ) X j (9)where ∆ ( T − v ) is the combinatorial Laplacian assigned to the graph T − v whichcan be obtained from the Laplacian ∆ associated to the triangulation T byremoving from its associated matrix the column and row labeled by the vertex v . The resulting amplitude is up to a factor given by Z (Σ T ) = e − µF ( T ) (cid:16) (2 π ) N ( V ) − Det ∆ ( T − v ) (cid:17) D (10)where N ( V ) − T − v , which is just thenumber of vertices of the triangulation T minus one and D is the dimension ofthe space-time in which our triangulated surface lives.The integral above in combinatorics is related to the Matrix-Tree theorem[10] where it is described that the determinant Det ∆ ( T − v ) equals the numberof spanning trees of the triangulation T . The Laplacian can be generalizedto a vertex or edge weights description where the above integral is generalized,however we do not describe it here.Observe that for a non-degenerate triangulation T , the number of spanningtrees is clearly greater than one. Therefore as the determinant appears as adenominator in the evaluation of the partition function it is clear that thepartition function evaluation is bounded from above as Z (Σ T ) < (cid:16) (2 π ) N ( V ) − (cid:17) D (11)which comes from a degenerate case in which it could exist only one tree.The partition function is given by the sum over all triangulations Z (Σ) = (cid:16) π (cid:17) D X T e − µF ( T ) (cid:16) (2 π ) N ( V ) Det ∆ ( T − v ) (cid:17) D (12)The question now is how do we perform the above sum over all triangulationsfor any arbitrary surface. This is what we do in section 4 by considering thegeneral orientable and non-orientable case, giving details of the calculations forsome examples. We also need to know how are all triangulations of a surfacegenerated. This is what we describe in the following section. See appendix for a description of the Matrix-Tree Theorem Graphs on surfaces: Triangulations
We first give some definitions. A graph is a pair G = ( V ( G ) , E ( G )) where V ( G ) = ∅ is called vertex set, and E ( G ) is a set where each element e ∈ E ( G )consist of a pair of elements of V ( G ). The elements of E ( G ) are called edges.Two vertices are said to be adjacent, if there is an element of E ( G ) which joinsthem. A graph with n vertices is complete and denoted K n if any two verticesare adjacent.A graph G is said to be embedded in a surface Σ if the vertices of G aredistinct points of Σ and every edge of G is a curve in Σ connecting the corre-sponding points.A triangulation of a surface Σ will be defined as an embedded graph T inthe surface such that Σ is divided into regions called faces, such that each faceis bounded by exactly three vertices and three edges, and any two faces haveeither one common vertex or one common edge or no common elements of thegraph.Two triangulations T and T are said to be isomorphic if there is a oneto one and onto mapping φ : V ( T ) −→ V ( T ) such that φ ( u ) φ ( v ) ∈ E ( T )whenever uv ∈ E ( T )Let T be a triangulation of a surface Σ, and consider an edge e and theirtwo triangles which contain it. Contract the edge e and replace the two doubleedges by single ones. This lead us to a new triangulation see fig[1]. The inversemove is called vertex splitting.Figure 1: Vertex splitting and edge contractionGiven a triangulation T of a surface Σ we can perform a vertex splittingor an edge contraction in order to obtain a new triangulation. When in atriangulation we cannot perform none edge contractions which lead to a newtriangulation again, we say that our triangulation is minimal.The sphere has only one 3-connected minimal triangulation given by theembedded graph K in the sphere [16], [19].And it is also known that all triangulations of the sphere are obtained fromthe singular minimal triangulation K [19] by vertex splittings.Now it is known that there are two minimal triangulations of the projectiveplane [6] one given by the embedding of K and the other given in figure[2]. Allthe triangulations of the projective plane are obtained from these two minimaltriangulations by vertex splittings. 6igure 2: Irreducible triangulations of the projective planeFinally, for the torus it was shown [15] that there are 21 minimal triangu-lations of it. For instance one is given by the embedding of K in the torus,15 triangulations with 8 vertices, 4 non-isomorphic ones with 9 vertices and 1irreducible one with 10 vertices, all of them non-isomorphic. And from these21 triangulations we can obtain all the triangulations of the torus by vertexsplittings moves.It is known that the set of minimal triangulations for every surface Σ is finite[7], [8] and the number grows rapidly.Given a graph G , a subgraph H is given by V ( H ) ⊆ V ( G ) and E ( F ) ⊆ E ( G ). When it happens that V ( H ) = V ( G ), H is called a spanning subgraph.A tree is a connected graph without cycles. Given a graph G we say that H is a spanning tree of G if H is a tree and a spanning graph.Given a graph G (or triangulation of a surface T ), it is also important toknow the number of spanning trees of it, as will be used in the next section.There is way to calculate the number of different spanning trees of the graphby the matrix-tree theorem given in the appendix.However, if we need to know how the number of spanning trees grows asthe triangulation of a surface has more and more vertices(as is needed for ourcalculations in the next section) the matrix-tree theorem is not very useful forthe purposes of computing the partition function.We therefore need to know a new way to calculate it which does not requiresuch a tedious calculation. Or we can try to give upper bounds for this number.This is what we do in the following section, we use an upper bound found in[13]. As a mathematical problem it will be interesting to have a better bound;or even better an exact way to describe it. In this section we compute our partition function following all the mathematicaldetails we described in our previous section. Our description is mathematicallyformal which gives a precision rule for doing any calculation for any surface.However it will be clear that even this combinatorial computations are farfrom being trivial and when the genus of the surface grows the computationsbecome so difficult and completely unknown. This is because we do not knowthe number of irreducible triangulations of all closed surfaces, and some studies7n this direction by finding upper bounds have been studied in [17] and recentlyapproached by [1].Our next step is to show the way to perform this computation. The thing isthat we can only give an approximation of it, and give an lower bound explicitlyfor the sphere only. Part of the calculation can also be given for the case ofthe torus. The combinatorial problem is complicated since as the number ofspanning trees grows when the triangulation has a larger number of vertices,the number of non-isomorphic triangulations with a certain number of verticesincreases a lot as well. In fact this latter problem is a very complicated one inthe field of combinatorics. We proceed now to our calculations. We denote asurface of genus g by S g . In our notation we denote the sphere by S . Generally we saw that the partitionfunction was given by equation (12). We are summing over triangulations, butit can be seen that such a sum can be translated into a series sum over integers,as we now show.As we have mentioned before, all of the triangulations of the sphere can beobtained by refining a single simple triangulation which is a minimal one [19].This minimal single triangulation of the sphere is given by the complete graph K , that is, the tetrahedron graph.In the language of topological graph theory we say that the complete graph K is embeddable in the sphere. This graph is our first summand of our partitionfunction. From this single triangulation we start taking vertex splittings.It is clear that the following summands are given when we take vertex split-tings over and over; observe that the number of faces is always even, that is, N ( F ) = n = 2 k , and the number of vertices is giving by N ( V ) = k + 2, whichcan be seen to be k + χ ( S ).This lead us to rewrite the partition function sum (12) as follows Z ( S ) = (cid:16) π (cid:17) D ∞ X k =2 e − µ k C ( T k +2 ) (cid:16) (2 π ) k +2 κ ( T k +2 ) (cid:17) D (13)where by C ( T k +2 ) we denote the number of non-isomorphic triangulations with N ( V ) = k + 2 vertices, and κ ( T k +2 ) denotes the number of spanning trees of atriangulation with N ( V ) = k + 2 vertices. For instance the first summand isgiven by only one single graph which is K where C ( T ) = 1 and κ ( T ) = 16.Each summand has contributions from the number of non-isomorphic trian-gulations with a fixed number of vertices and from the number of trees of thistriangulations.The number of non-isomorphic triangulations of the sphere with a fixednumber of vertices has an asymptotic behavior [20]. This number is giving by8 ( T k +2 ) ∼ (cid:16) π (cid:17) ( k + 2) − (cid:16) (cid:17) k +3 (14)The number of spanning trees for two non-isomorphic triangulations T and T with the same number of vertices( N ( V ) = k + 2), are different since their Tuttepolynomial invariant is different . For this reason, the calculation is harderthan thought. The only thing we can do now is to use an upper bound for thenumber of trees on any triangulation with N ( V ) = k + 2 vertices. As proved in[13] any triangulation with k + 2 vertices has an upper bound for the numberof spanning trees given by κ ( T k +2 ) ≤ k + 2 (cid:16) k + 2) k + 1 (cid:17) k +1 (15)We therefore have Z ( S ) ≥ (cid:16) π (cid:17) D (cid:16) π (cid:17) ∞ X k =2 e − µ k ( k +2) − (cid:16) (cid:17) k +3 (cid:16) (2 π ) k +2 ( k + 2)( k + 1) k +1 (3( k + 2)) k +1 (cid:17) D (16)which tells us that we have a lower bound. It is now a task to study its con-vergence for values of the parameter µ and of the dimension D . It can be seenthat the above partition function converges for any value of µ ≥ µ = 2, for D = 1 we have Z ( S ) ≥ . D = 2 Z ( S ) ≥ . D , the partition function goes to infinity,but also we have that for larger values of µ the partition function convergesmore rapidly. We therefore have that for a fixed value of the dimension D , thepartition function always converges for µ ≥ As in the case of the sphere, all the triangulations of the torus can be obtainedby refining the minimal triangulations of it. In the case of the sphere we hadonly one minimal triangulation. For the case of the torus we have 21 non-isomorphic minimal triangulations [15] from which we start in order to obtainall of the remaining ones.For instance the torus S has as its simpler triangulation one given by theembedding of the complete graph K . Therefore it has 7 vertices, 21 edgesand 14 faces. We also have 15 non-isomorphic triangulations with 8 vertices, 4non-isomorphic ones with 9 vertices and 1 irreducible one with 10 vertices. The number of spanning trees in a graph is a special case of the Tutte Polynomial(seeapendix)
9n all of these triangulations we can see that the number of faces is alwayseven, that is, N ( F ) = n = 2 k ; we also have that N ( V ) = k = k + χ ( S ). Thisleads to the following sum Z ( S ) = (cid:16) π (cid:17) D ∞ X k =7 e − µ k C ( T k ) (cid:16) (2 π ) k κ ( T k ) (cid:17) D (17)where again C ( T k ) denotes the number of non-isomorphic triangulations with N ( V ) = k vertices for the torus, and κ ( T k ) is the number of spanning trees ofa triangulation graph with k vertices. The upper bound number of spanningtrees is the same we used before for the sphere since it is just a number whichdepends on the number of vertices of the graph. But now our problem is thatthe number of non-isomorphic triangulations C ( T k ) of the torus is not known inany way. It is just as simple as noticing that we now have C ( T ) = 1 , C ( T ) =15 , C ( T ) = 4 , C ( T ) = 1. Then the sum above can be taken to the followingexpression Z ( S ) = (cid:16) π (cid:17) D h e − µ (cid:16) (2 π ) κ ( T ) (cid:17) D + 5 e − µ (cid:16) (2 π ) κ ( T ) (cid:17) D + 20 e − µ (cid:16) (2 π ) κ ( T ) (cid:17) D +21 ∞ X k =10 e − µ k C ( T k ) (cid:16) (2 π ) k κ ( T k ) (cid:17) D i (18)where the major contribution is obviously given by Z ( S ) ∼ (cid:16) π (cid:17) D ∞ X k =10 e − µ k C ( T k ) (cid:16) (2 π ) k κ ( T k ) (cid:17) D (19)The real thing is that if we do not know anything about the number C ( T k ),except for the irreducible triangulations, we cannot compare the torus partitionfunction to the sphere one.However we would like to show only a partial comparison. This partialcomparison will be done by considering that there is only one triangulationwith a fixed number of vertices, for the sphere and for the torus.Suppose then that there is only one triangulation for the sphere with k + 2vertices, that is C ( T k +2 ) = 1. Then Z ( S ) partial ∼ (cid:16) π (cid:17) D ∞ X k =2 e − µ k (cid:16) (2 π ) k +2 κ ( T k +2 ) (cid:17) D (20)For the torus we have that there are 21 irreducible triangulations from whichwe generate all triangulations. Suppose then that each of the 21 irreducibletriangulations generate only one respective class of triangulations with a fixednumber of vertices. We write 10 ( S ) partial ∼ (cid:16) π (cid:17) D ∞ X k =10 e − µ k (cid:16) (2 π ) k κ ( T k ) (cid:17) D (21)both sums are partial but they still contain a sum over a very large number oftriangulations. The thing is that if we take µ ≥ D we have that Z ( S ) partial ≫ Z ( S ) partial (22)The above inequality is a very strict one and it tells us that the partial contri-bution of the sphere is really much more bigger than the partial contributionof the torus. Of course this is not telling us that the original sums obey thesame inequality, but the interesting thing is the following. The number of non-isomorphic triangulations with a fixed number of vertices for the torus, is biggerthan the one for the sphere with the same number of fixed vertices. We havealso mentioned that this number grows exponentially when the genus of thesurface grows. Therefore it is expected that the inequality (22) changes whenconsidering the complete calculation.It can also be suggested that partial contributions from other topologicalsurfaces are also dominated by the lowest genus surface.Let us now give the partition function sum expression for any orientableclosed surface.Observe first the following, which we assume happens for all of the differenttopologies: The sums for the sphere and the partial sum of the torus show that inthe summands 2 π has exponent k + χ (Σ), where χ (Σ) is the Euler characteristicof the surface. We also have a factor which multiplies the summand given bythe number of non-isomorphic irreducible triangulations of the surface, whichfor the sphere it was one and for the torus it was 21. The sum starts also froma higher number when the genus of the surface increases. For the torus it startsfor k = 10 where 10 is the number of vertices of the irreducible triangulationwith more vertices. The sum for any surfaces of any genus is given by Z ( S g ) = (cid:16) π (cid:17) D ∞ X k = n e − µ k ( C T k + χ ( Sg ) ) (cid:16) (2 π ) k + χ ( S g ) κ ( T k + χ ( S g ) ) (cid:17) D (23)where again we have that C ( T k + χ ( S g ) ) denotes the number of non-isomorphictriangulations with N ( V ) = k + χ ( S g ) vertices for the surface of genus χ ( S g ),and κ ( T k + χ ( S g ) ) is the number of spanning trees of a triangulation graph with k + χ ( S g ) vertices. With the calculation of the sphere and the way we explained how the sumfor the torus and any surface is to be obtained we could easily know how the11alculations follows for any non-orientable surface. The only difference wouldbe the appearance of the non-orientable euler characteristic.For instance recall that the projective plane has two irreducible triangula-tions from which we can obtain all of its triangulations by the vertex splittingmoves. One is given by the embedding of the complete graph K , with 6 vertices,15 edges and 10 faces. Then each triangulation obtained from this irreducibleone, by the splitting moves, will have an even number of faces 2 k and k + 1 ver-tices where k starts from 5. The second irreducible triangulation has 7 vertices,18 edges and 12 faces, and all of the triangulations obtained from this irreducibleone, will have also an even number of faces 2 k and k + 1 vertices where k starsfrom 6. Denote the projective plane by N . Therefore the partition function isgiven by Z ( N ) = (cid:16) π (cid:17) D h e − µ (cid:16) (2 π ) κ ( T ) (cid:17) D + 2 ∞ X k =6 e − µ k C ( T k +1 ) (cid:16) (2 π ) k +1 κ ( T k +1 ) (cid:17) D i (24)We can easily guess and generalize the above sum to any non-orientable surfaceof genus g . Denote such surface by N g . We therefore have the generalizedpartition function given by Z ( N g ) = (cid:16) π (cid:17) D ∞ X k = n e − µ k ( C T k + χ ( Ng ) ) (cid:16) (2 π ) k + χ ( N g ) κ ( T k + χ ( N g ) ) (cid:17) D (25)where as for the orientable case C ( T k + χ ( N g ) ) denotes the number of non-isomorphictriangulations with N ( V ) = k + χ ( N g ) vertices for the non-orientable surface ofgenus χ ( S g ), and κ ( T k + χ ( N g ) ) is the number of spanning trees of a triangulationgraph with k + χ ( N g ) vertices. We have seen in this paper that there is a need to understand deeper a puremathematics problem in order to have a complete calculation of the partitionfunction of any two dimensional surface. In order to have a complete sumover all triangulations of a surface we learnt that we need to know first all thenon-isomorphic irreducible triangulations of the surface.The problem clearly would be to have an asymptotic expression for thenumber of non-isomorphic triangulations of any surface. Until now, we havethis expression for the sphere only [20]. And it is even very hard to find atleast the number of irreducible triangulations of a surface. There have beenonly upper bounds for the number of irreducible triangulations of a surface ofgenus χ ( S g ) [1], [17]. Even finding non-isomorphic complete graph orientable ornon-orientable embeddings of complete graphs on surfaces gives a huge numberof families [5], [11], [14]. 12herefore the problem of computing partition functions for any surface isincomplete. We therefore have that the discrete formulation which we pre-sented here, is not an advantage over the continuous evaluations. It will be anadvantage if we first solve the combinatorial problems we presented here. ACKNOWLEDGMENTS
I want to thank Isidoro Gitler for very useful conversations related to com-binatorics and for pointing me to references.
A The spanning trees of a triangulation
This appendix describes the matrix-tree theorem. This is in order to just un-derstand how it is used in the paper. For a deeper description of it see [9].Let G denote a connected graph with vertex set V ( G ) and edges set E ( G ).The combinatorial Laplacian ∆ G for the graph G is defined in section 2, andit is given by a square matrix indexed by their vertices. This square matrixis completely symmetric and has determinant zero. Given any vertex v of G consider the cofactor ∆ G − v of the matrix Laplacian ∆ G given by deleting from∆ G the row and column indexed by the vertex v . Matrix-Tree Theorem . The determinant
Det (∆ G − v ) is independent ofthe vertex v and equals the number of spanning trees of G .There is also a generalization of the matrix-tree theorem when consideringgraphs with edge weights. The number of spanning trees of a graph can bethought as an invariant of the graph. This is because this number is a particularcase of a more general invariant associated to graphs via a polynomial discoveredby Tutte [21].The Tutte polynomial of a graph is a two variable one T ( G ; x, y ) which isdefined by the contraction-deletion rule.1.- If G has no edges then T ( G ; x, y ) = 12.- T ( G ; x, y ) = T ( G − e ; x, y ) + T ( G \ e ; x, y ) where e is neither a loop nora bridge and G − e and G \ e denote the result of deleting and contracting theedge e .3.- T ( G ; x, y ) = yT ( G − e ; x, y ) when e is a loop4.- T ( G ; x, y ) = xT ( G/e ; x, y ) when e is a bridgeThis are the properties which define the Tutte Polynomial. It happens thatwhen x = 1 , y = 1, the Tutte polynomial of the graph G gives the number ofits spanning trees. 13 eferences [1] G.Aguilar Cruz, F.Zaragoza Mart´ınez, An upper bound on the size of irre-ducible triangulations, Draft (2006)[2] J.Ambjorn, B.Durhuus, J.Frohlich, P.Orland, The appearance of criticaldimensions in regulated string theories, Nuclear Phys.B , (1986) 457-482[3] J.Ambjorn, B.Durhuus, J.Frohlich, P.Orland, The appearance of criticaldimensions in regulated string theories II, Nuclear Phys.B , (1986)161-184[4] J. Ambjorn, B. Durhuus, T. Jonsson, Quantum Geometry: A statisticalfield theory approach, Cambridge University Press (1997)[5] J.L.Arocha, J.Bracho, V.Neumann-Lara, Tight and untight triangulationsof surfaces by complete graphs, Journal of Combinatorial Theory Series B , (1982) 222-230[7] D.Barnette, A.Edelson, All orientable 2-manifolds have finitely many min-imal triangulations, Israel J.Math (1996) 316-327[11] L.Goddyn, L.B.Richter, J.Siran, Triangular embeddings of complete graphsfrom graceful labellings of paths, Preprint submitted to Elsevier Science(2006)[12] M.Green, J.Schwarz, E.Witten, Superstring theory, two volumes, Cam-bridge University Press, (1987)[13] G.R.Grimmett, An upper bound for the number of spanning trees of agraph, Discrete Math, (1976) 323-324[14] V.P.Korzhik, H.J.Heinz-Jrgen Voss, Exponential families of nonisomorphicnonorientable genus embeddings of complete graphs, Journal of Combina-torial Theory, Series B , (1987) 52-62[16] B.Mohar, C.Thomassen, Graphs on surfaces, The Johns Hopkins Univer-sity Press, (2001)[17] A.Nakamoto, K.Ota, Note on irreducible triangulations of surfaces,J.Graph Theory, , (1995) 227-233[18] J.Polchinski, String Theory, two volumes, Cambridge University Press,(1998)[19] E.Steinitz, H.Rademacher, Vorlesunger uber die Theorie der Polyeder,Springer, Berlin (1934)[20] W.T.Tutte, A census of planar triangulations, Canad. J. Math. (1962),21-38.[21] W.T.Tutte, A contribution to the theory of chromatic polynomials, Cana-dian J.Math.6