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Graphs on surfaces is an active topic of pure mathematics belonging to graphtheory. It has also been applied to physics and relates discrete and continuousmathematics. In this paper we present a formal mathematical description of therelation between graph theory and the mathematical physics of discrete stringtheory. In this description we present problems of the combinatorial world ofreal importance for graph theorists. The mathematical details of the paper areas follows: There is a combinatorial description of the partition function ofbosonic string theory. In this combinatorial description the string world sheetis thought as simplicial and it is considered as a combinatorial graph. It canalso be said that we have embeddings of graphs in closed surfaces. The discretepartition function which results from this procedure gives a sum overtriangulations of closed surfaces. This is known as the vacuum partitionfunction. The precise calculation of the partition function depends oncombinatorial calculations involving counting all non-isomorphic triangulationsand all spanning trees of a graph. The exact computation of the partitionfunction turns out to be very complicated, however we show the exactexpressions for its computation for the case of any closed orientable surface.We present a clear computation for the sphere and the way it is done for thetorus, and for the non-orientable case of the projective plane.

Graphs on Surfaces and the Partition Function of String Theory