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So's conjecture for integral circulant graphs of $4$ types

Hao Li, Xiaogang Liu

Abstract

In [Discrete Mathematics 306 (2005) 153-158], So proposed a conjecture saying that integral circulant graphs with different connection sets have different spectra. This conjecture is still open. We prove that this conjecture holds for integral circulant graphs whose orders have prime factorization of $4$ types.

So's conjecture for integral circulant graphs of $4$ types

Abstract

In [Discrete Mathematics 306 (2005) 153-158], So proposed a conjecture saying that integral circulant graphs with different connection sets have different spectra. This conjecture is still open. We prove that this conjecture holds for integral circulant graphs whose orders have prime factorization of types.
Paper Structure (11 sections, 39 theorems, 112 equations, 4 tables)

This paper contains 11 sections, 39 theorems, 112 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main']}
  4. (a) and (b) of Theorem \ref{['main']}
  5. Useful notations and lemmas
  6. Proofs of (a) and (b) of Theorem \ref{['main']}
  7. (c) and (d) of Theorem \ref{['main']}
  8. Useful notations and lemmas
  9. Proof of (c) of Theorem \ref{['main']}
  10. Proof of (d) of Theorem \ref{['main']}
  11. Conclusion

Key Result

Lemma 1.1

(Seeso0) A circulant graph $\mathrm{Cay}(\mathbb{Z}_{n},S)$ is integral if and only if $S$ is a union of $G_{n}(d)$'s for some divisors $d$ of $n$.

Theorems & Definitions (73)

  • Conjecture 1
  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 63 more