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

Josué I. Rios-Cangas

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.

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

Abstract

We deal in this work with a class of graphs, namely, the class of distance-regular graphs, in which on the basis of -adjacency operators, the adjacency operator of a distance-regular graph is identified as a Jacobi matrix. To get so, the set of the -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 . 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.

Paper Structure

This paper contains 7 sections, 15 theorems, 79 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bigraphs and the $k$-adjacency operators
  3. Distance-regular graphs and the adjacency Jacobi operator
  4. Distance-regular graphs with finite diameter
  5. Examples
  6. Regular trees
  7. Complete graphs

Key Result

Proposition \oldthetheorem

A graph is bipartite if and only if it has no cycles of odd length.

Figures (5)

  • Figure 1: Infinite star graph and grid graph are bipartite.
  • Figure 2: Eigenvalues of $J_\tau$.
  • Figure 3: $T_2$ and $T_3$.
  • Figure 4: $K_3$, $K_4$ and $K_6$.
  • Figure 5: Eigenvalues \ref{['eq:eigenvalue-function']} vs. $J_{n-2}$.

Theorems & Definitions (49)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 39 more