Topological properties of generalized Markoff mod p graphs

2025/12/26 by Shohei Satake, Satake, Shohei, Yoshinori Yamasaki +1 · 1 citation
Computer Science · #Advanced Graph Theory Research #Topological and Geometric Data Analysis #Computational Geometry and Mesh Generation

paper · doi:10.48550/arxiv.2512.21963

Abstract

The generalized Markoff mod p graph is defined via the equation x²+y²+z²=xyz+κ over the finite field \mathbbF_p of prime order p. In this paper, we investigate the topological properties of the graph such as non-planarity, surface embeddability, and the existence of short cycles. Our approach is based on a systematic construction of K_3,3-subdivisions, integrating techniques from graph theory, computer algebra, and number theory.

Citations

• Divisibility by p for Markoff-like Surfaces
• A new proof of Chen's theorem for Markoff graphs
• An almost linear time algorithm testing whether the Markoff graph modulo p is connected
• Connectivity of Markoff mod-p graphs and maximal divisors
• A Heuristic Subexponential Algorithm to Find Paths in Markoff Graphs Over Finite Fields
• A Cryptographic Hash Function from Markoff Triples
• Non-planarity of Markoff graphs mod p
• Chebyshev-Fibonacci polynomial relations using generating functions
• Nonabelian level structures, Nielsen equivalence, and Markoff triples
• Experiments with the Markoff surface
• Markoff Triples and Strong Approximation
• Expander Graphs in Pure and Applied Mathematics

Cited by

• Counting in Vieta graphs over \mathbbF_p

Related

• Non-planarity of Markoff graphs mod p
• Connectedness of special points in the Markoff mod p graphs
• Connectivity of Markoff mod-p graphs and maximal divisors
• A new proof of Chen's theorem for Markoff graphs
• Bounding Lifts of Markoff Triples mod p