  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.
