The k-adjacency operators and adjacency Jacobi matrix on distance-regular graphs

We deal in this work with a class of graphs, namely, the class of distance-regular graphs, in which on the basis of k-adjacency operators, the adjacency operator A of a distance-regular graph is identified as a Jacobi matrix. To get so, the set of the k-adjacency operators is recognized as a canonical basis in a certain Hilbert space, where the spectrum of the Jacobi matrix coincides with the support of the measure of A. The obtained identification permits a deeper spectral analysis of the graph. The finite-dimensional case is addressed by means of the extension theory of nondensely defined, symmetric linear operators.