Enumeration of Triangles in Residue Graphs
Faculty Mentor
Joshua Lambert
Location
Ogeechee Theater
Type of Research
Completed
Session Format
Oral Presentation
College
Allen E. Paulson College of Engineering & Computing
Department
Computer science
Abstract
Let p be a odd prime satisfying p ≡ 1 (mod 2t+1), ensuring that the set of 2t residue forms a symmetric subset of F× p . We construct a 2t residue graph whose vertices correspond to the elements of Fp, with an edge between x, y ∈ Fp whenever x − y ∈ F×2t p . The enumeration of triangles in these graphs is reduced to counting the number of consecutive 2t residues in Fp, which determines the frequency of solutions to the equation x + y + z = 0 within the residue class. We focus on the quartic case (t = 2) to illustrate the increased complexity relative to the quadratic case. Using multiplicative characters and Jacobi sums, we derive closed-form expressions for triangle counts. The goal is to establish a systematic framework for computing triangle counts in 2t residue graphs and to demonstrate how algebraic properties of primes govern their combinatorial structure.
Program Description
.
Start Date
4-21-2026 11:30 AM
End Date
4-21-2026 11:45 AM
Recommended Citation
Okoye, Chukwukelum Williams, "Enumeration of Triangles in Residue Graphs" (2026). GS4 Student Scholars Symposium. 50.
https://digitalcommons.georgiasouthern.edu/research_symposium/2026A/2026A/50
Enumeration of Triangles in Residue Graphs
Ogeechee Theater
Let p be a odd prime satisfying p ≡ 1 (mod 2t+1), ensuring that the set of 2t residue forms a symmetric subset of F× p . We construct a 2t residue graph whose vertices correspond to the elements of Fp, with an edge between x, y ∈ Fp whenever x − y ∈ F×2t p . The enumeration of triangles in these graphs is reduced to counting the number of consecutive 2t residues in Fp, which determines the frequency of solutions to the equation x + y + z = 0 within the residue class. We focus on the quartic case (t = 2) to illustrate the increased complexity relative to the quadratic case. Using multiplicative characters and Jacobi sums, we derive closed-form expressions for triangle counts. The goal is to establish a systematic framework for computing triangle counts in 2t residue graphs and to demonstrate how algebraic properties of primes govern their combinatorial structure.
