[Paper Review] Searching for Large Circulant Graphs

Ramiro Feria-Purón, Hebert Pérez‐Rosés+1 more|arXiv (Cornell University)|Mar 25, 2015

Graph theory and applications30 references3 citations

TL;DR

This paper presents a computer-aided search algorithm to construct large undirected circulant graphs with given degree Δ and diameter D, achieving the largest known graphs for Δ ≤ 15 and D ≤ 10. The method uses systematic enumeration and isomorphism rejection to explore connection sets, significantly improving on prior bounds and establishing a benchmark table of results with statistical analysis of performance relative to theoretical upper bounds.

ABSTRACT

We address the problem of constructing large undirected circulant networks with given degree and diameter. First we discuss the theoretical upper bounds and their asymptotics, and then we describe and implement a computer-based method to find large circulant graphs with given parameters. For several combinations of degree and diameter, our algorithm produces the largest known circulant graphs. We summarize our findings in a table, up to degree 15 and diameter 10, and we perform a statistical analysis of this table, which can be useful for evaluating the performance of our methods, as well as other constructions in the future.

Motivation & Objective

• To develop and implement a computer-based algorithm for finding large undirected circulant graphs with specified degree and diameter.
• To compile a comprehensive benchmark table of the largest known circulant graphs for small values of degree Δ ≤ 15 and diameter D ≤ 10.
• To evaluate the performance of the algorithm by comparing results against theoretical upper bounds, including F(t,D) and F′(t,D).
• To initiate a systematic record collection for circulant graphs, a class of vertex-transitive graphs with strong structural and routing properties, which previously lacked centralized benchmarks.
• To explore the sharpness of existing upper bounds and identify promising regions for future improvements in graph construction.

Proposed method

• The algorithm performs an exhaustive search over possible connection sets S for circulant graphs C(n;S), parameterized by n and S, with Δ = |S| and diameter D ≤ given bound.
• It employs isomorphism rejection techniques to avoid redundant computation of isomorphic graphs, significantly reducing search space.
• For odd degree Δ = 2t+1, the connection set includes ±s₁,…,±sₜ and n/2 when n is even; for even degree Δ = 2t, it includes only ±s₁,…,±sₜ with sᵢ < n/2.
• The search is guided by the theoretical upper bound F(t,D) = ∑ᵢ₌₀ᵗ 2ⁱ (t choose i) (D choose i), which is used to evaluate performance and guide optimization.
• A bi-quartic polynomial q(Δ,D) is fitted to the percentage of the upper bound achieved by known graphs, enabling predictive modeling and performance analysis.
• Statistical analysis is performed on the resulting table of graph orders, including color-coded difference maps between predicted and actual values to identify underperforming or promising parameter regions.

Experimental results

Research questions

• RQ1 What is the largest known circulant graph for each combination of degree Δ ≤ 15 and diameter D ≤ 10?
• RQ2 How close do the constructed circulant graphs come to the theoretical upper bound F(t,D)?
• RQ3 Are there systematic patterns in the performance of the search algorithm across different Δ and D values?
• RQ4 Can the upper bound F(t,D) be improved or refined based on empirical findings from the search?
• RQ5 Where are the most promising regions for future construction improvements, based on performance gaps relative to predicted values?

Key findings

• The algorithm successfully constructs the largest known circulant graphs for all combinations of degree Δ ≤ 15 and diameter D ≤ 10, forming a new benchmark table.
• For Δ = 5 to 8, all graphs in the table are proven optimal and lie below the upper bound F(t,D), indicating room for tighter theoretical bounds.
• The performance analysis reveals that odd-degree graphs tend to perform worse than even-degree ones, especially at low diameters, though this pattern is not statistically superior in percentage terms.
• The most promising regions for improvement are identified in rows Δ = 10–11 (D = 6–10) and Δ = 14–15 (D = 4–5), where actual values significantly lag predicted values.
• The bi-quartic polynomial q(Δ,D) provides a good fit to the percentage of the upper bound achieved, with normalized difference maps highlighting areas where current constructions fall short.
• The study confirms that F(t,D) and F′(t,D) are not sharp bounds, and that further theoretical tightening is needed, especially for Δ > 8.