[Paper Review] Isomorphic and Nonisomorphic, Isospectral Circulant Graphs

Julia M. Nowlin Brown|ArXiv.org|Apr 13, 2009

Finite Group Theory Research11 references3 citations

TL;DR

This paper presents a new spectral characterization condition for circulant graphs: if a circulant graph of order $ n = p_1^{r_1} \cdots p_s^{r_s} $ has connection set size $ m $, and $ p_1 \geq m $ with either $ s=1 $ or $ p_2 > p_1(m-1) $, then it is uniquely determined up to isomorphism by its spectrum. Additionally, it constructs infinite families of isospectral, non-isomorphic circulant graphs of order $ 2^r p $ for odd prime $ p $ and $ r > 2 $, using symmetric connection sets derived from arithmetic progressions modulo $ n $, proving non-isomorphism via group ring support and number-theoretic properties.

ABSTRACT

New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley graphs of cyclic groups with the same list of eigenvalues of their adjacency matrices will be presented.

Motivation & Objective

• To identify conditions under which circulant graphs are completely determined by their adjacency matrix spectra.
• To resolve the gap in knowledge about when isospectral circulant graphs must be isomorphic.
• To construct explicit families of isospectral, non-isomorphic circulant graphs, particularly for orders $ 2^r p $ with $ r > 2 $.
• To extend existing constructions beyond dihedral and PSL groups to a broader class of abelian groups.
• To address the open problem of isomorphism status for extended constructions with multiple connection sets.

Proposed method

• Use the eigenvalue formula for circulant graphs: $ \lambda_x = \sum_{s \in S} \omega^{xs} $, where $ \omega $ is a primitive $ n $th root of unity.
• Apply group ring techniques in $ \mathbb{Z}[\mathbb{Z}_n] $, mapping group elements to roots of unity via $ \varphi(z) = \omega $, and analyze support and coefficient structures.
• Define connection sets $ A $ and $ B $ as unions of arithmetic progressions modulo $ n = 2^r p $, ensuring equal spectral sums via symmetry and modular arithmetic.
• Prove non-isomorphism by showing that one set contains elements divisible by $ p $, while the other does not, violating equivalence under multiplication by units modulo $ n $.
• Extend the construction by taking $ \tilde{A} = A \cup qA $, $ \tilde{B} = B \cup qB $ for $ q $ coprime to $ n $, preserving spectral equality.
• Use Musychuk’s result on Ádám’s conjecture to confirm non-isomorphism when $ n = 4p $, and leave the general case open.

Experimental results

Research questions

• RQ1 Under what conditions on the order $ n $ and connection set size $ m $ is a circulant graph uniquely determined by its spectrum?
• RQ2 Can isospectral, non-isomorphic circulant graphs be systematically constructed for abelian groups beyond known examples?
• RQ3 Do the extended constructions $ \tilde{A} = A \cup qA $, $ \tilde{B} = B \cup qB $ yield isomorphic graphs for $ r > 2 $?
• RQ4 How do number-theoretic properties of connection sets (e.g., divisibility by $ p $) affect isomorphism in circulant graphs?
• RQ5 Is there a spectral or algebraic invariant that can distinguish the extended graphs $ \tilde{X} $ and $ \tilde{Y} $ when $ r > 2 $?

Key findings

• The paper establishes a new spectral characterization: if $ n = p_1^{r_1} \cdots p_s^{r_s} $ with $ p_1 \geq m $ and either $ s=1 $ or $ p_2 > p_1(m-1) $, then isospectral circulant graphs must be isomorphic.
• For $ n = 2^r p $ with $ r > 2 $, the construction yields isospectral, non-isomorphic circulant graphs using connection sets $ A $ and $ B $, each of size $ 2p $, derived from arithmetic progressions modulo $ n $.
• The connection sets $ A $ and $ B $ are not equivalent under multiplication by units modulo $ n $, and one contains elements divisible by $ p $, while the other does not, proving non-isomorphism.
• The extended graphs $ \tilde{X} = \text{Cay}(\mathbb{Z}_n, \tilde{A}) $, $ \tilde{Y} = \text{Cay}(\mathbb{Z}_n, \tilde{B}) $ have identical spectra due to eigenvalue decomposition: $ \tilde{\lambda}_x = \lambda_x + \lambda_{qx} $.
• For $ n = 4p $, the graphs are non-isomorphic due to Musychuk’s result on Ádám’s conjecture, as one connection set contains elements divisible by $ p $, the other does not.
• The isomorphism status of the extended graphs for $ r > 2 $ remains open, posing a key unresolved problem in the paper.