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On $U$-unitary Cayley graphs over finite rings

Tung T. Nguyen, Nguyen Duy Tân

Abstract

Graphs defined over a finite ring are well-studied in the literature. Due to their nature, these types of graphs connect several branches of mathematics, including algebra, number theory, matrix theory, and representation theory. In recent work, we studied $U$-unitary Cayley graphs over a finite commutative ring, which unifies several constructions of graphs with arithmetic origins. Among various structural graph-theoretic results on these graphs--such as their connectedness, primeness, and bipartiteness--we show that their spectra can be described via a certain supercharacter theory. Utilizing this spectral description, we are able to find some classes of gcd-graphs that possess perfect state transfer. In this article, we generalize this study to finite non-commutative rings, with a special focus on the case of the matrix rings with coefficients in a finite field. We show, in particular, that gcd-graphs over these matrix rings have no perfect state transfer.

On $U$-unitary Cayley graphs over finite rings

Abstract

Graphs defined over a finite ring are well-studied in the literature. Due to their nature, these types of graphs connect several branches of mathematics, including algebra, number theory, matrix theory, and representation theory. In recent work, we studied -unitary Cayley graphs over a finite commutative ring, which unifies several constructions of graphs with arithmetic origins. Among various structural graph-theoretic results on these graphs--such as their connectedness, primeness, and bipartiteness--we show that their spectra can be described via a certain supercharacter theory. Utilizing this spectral description, we are able to find some classes of gcd-graphs that possess perfect state transfer. In this article, we generalize this study to finite non-commutative rings, with a special focus on the case of the matrix rings with coefficients in a finite field. We show, in particular, that gcd-graphs over these matrix rings have no perfect state transfer.
Paper Structure (14 sections, 27 theorems, 35 equations)

This paper contains 14 sections, 27 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Outline.
  3. $U$-unitary Cayley graphs and their graph theoretic properties
  4. $U$-unitary Cayley graphs
  5. Graph-theoretic properties of $U$-unitary Cayley graphs
  6. Connectedness
  7. Primeness
  8. Unitary Cayley graphs
  9. Primeness and singularity of the adjacency matrix
  10. Spectra theory of $U$-unitary Cayley graphs over a finite symmetric Frobenius algebra
  11. Symmetric Frobenius algebras
  12. Spectral description of $U$-unitary Cayley graphs
  13. The case of matrix rings
  14. Perfect state transfer on $U$-unitary Cayley graphs

Key Result

Proposition 2.5

The following conditions are equivalent.

Theorems & Definitions (62)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Corollary 2.6
  • proof
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • ...and 52 more