A simple computer-oriented method is presented for constructing the (molecular) distance matrix. The distance matrix considered is the graph-theoretical (topological) distance matrix.Chemical or molecular graphs, adequately labeled, may be associated with several matrices,l sometimes called topological matrices.2 Chemical or molecular graphs provide a pictorial representation of the connectivity in a molecule, with vertices (points) corresponding to individual atoms and edges (lines) depicting the valence bonds between pairs of atoms.3 In order to simplify the handling of chemical graphs we will use hydrogensuppressed graph^,^ i.e., chemical graphs depicting the skeletons of molecules pruned of their hydrogen atoms. A graph G is labelled if the vertices in G are numbered.Among the graph-theoretical (topological) matrices the most popular is the adjacency m a t r i~,~.~.~ but the distance matrix has also found a wide range of applications.6-26 In the recent article by R o~v r a y~~ the uses of the distance matrix in chemistry and other sciences (e.g., the physical, biological, and social sciences) are well documented. The same author has also shown that the distance matrix has been employed in disciplines relatively remote from chemistry such as anthropology, geography, geology, ornithology, philology, psychology, etc.The distance matrix in chemistry has been based on the geometric distances and on the graph-theoretical (topological) distances.27 Graph-theoretical (topological) distance is the length of the shortest path (i.e., the minimum number of edges) between the ith vertex and the jth vertex of a graph. Our interest in this work is focused on the graph-theoretical distance matrix. The uses of the distance matrix based on geometric distances are discussed by Crippen.28 In the present work whenever we mention the distance matrix, we will always refer to the graph-theoretical (topological) distance matrix.
Abstract. Let f be a non-holomorphic automorphic form of real weight and eigenvalue ,~ = 88 -p2, 9tp/> 0, which is defined with respect to a Fuchsian group of the first kind. Assume that oo is a cusp of this group and denote by a .... n ~ ~, the Fourier coefficients off at oe. Following Hecke and Maas we prove that under suitable assumptions the associated Dirichlet series L + (f, s) = ~, > 0 a| (n +/to)-' and L-(f, s) = = ~n < 0 a~,n In +/~l-' have meromorphic continuation in the entire complex plane and satisfy a certain functional equation ~ denotes the cusp parameter of the cusp oe). We are interested in mean square estimates of these functions. Iff is not a cusp form we
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