Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Logical aspects of isomorphism of controllable graphs and cospectrality of distance-regularized graphs

Aida Abiad, Anuj Dawar, Octavio B. Zapata-Fonseca

Abstract

We consider isomorphism of controllable graphs and cospectrality of distance-regularized graphs (which are known to be distance-regular or distance-biregular) in relation to logical definability. While most characterizations of these equivalence relations for such graph classes are of algebraic and spectral flavor, here we inject tools from first-order logic, extending and unifying several existing results.

Logical aspects of isomorphism of controllable graphs and cospectrality of distance-regularized graphs

Abstract

We consider isomorphism of controllable graphs and cospectrality of distance-regularized graphs (which are known to be distance-regular or distance-biregular) in relation to logical definability. While most characterizations of these equivalence relations for such graph classes are of algebraic and spectral flavor, here we inject tools from first-order logic, extending and unifying several existing results.
Paper Structure (19 sections, 17 theorems, 44 equations, 3 figures)

This paper contains 19 sections, 17 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First-order logic of graphs
  4. Iterated degree sequences
  5. Fractional isomorphism
  6. Graph spectra
  7. Coherent configurations
  8. The coherent configuration of a graph
  9. Intersection equivalence
  10. Isomorphism of controllable graphs
  11. Controllable graphs
  12. Walk-equivalence
  13. Isomorphism and $C^2$-equivalence
  14. Cospectrality of distance-regularized graphs
  15. Distance-regularized graphs
  16. ...and 4 more sections

Key Result

Theorem 2.1

Two graphs are $C^2$-equivalent if and only if they have equal iterated degree sequences.

Figures (3)

  • Figure 1: Coherent configuration of rank $5$ determined by the $8$-cycle.
  • Figure 2: A distance-biregular graph which is not distance-regular.
  • Figure 3: Coherent configuration of rank $9$ determined by the subdivision of the complete graph $K_4$.

Theorems & Definitions (23)

  • Theorem 2.1: IL
  • Theorem 2.2: RSU
  • Theorem 2.3: DSZ
  • Theorem 2.4: CFI
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1: G
  • Theorem 3.2
  • ...and 13 more