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

TL;DR

This paper explores the spectral properties of distance-regular graphs by representing their adjacency operators as Jacobi matrices using $k$-adjacency operators, enabling deeper spectral analysis.

Contribution

It introduces a novel representation of adjacency operators as Jacobi matrices via $k$-adjacency operators in distance-regular graphs, linking graph theory with spectral analysis.

Findings

Adjacency operator identified as a Jacobi matrix.

Spectrum coincides with the support of the measure of $A$.

Extension theory applied to finite-dimensional cases.

Abstract

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.