Some Necessary and Sufficient Conditions for Diophantine Graphs

2024/12/29 by M. A. Seoud, Seoud, M. A., A. Elsonbaty +5
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2412.20562

openalex publication_date 2024/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

A linear Diophantine equation ax + by = n is solvable if and only if gcd(a; b) divides n. A graph G of order n is called Diophantine if there exists a labeling function f of vertices such that gcd(f(u); f(v)) divides n for every two adjacent vertices u; v in G. In this work, maximal Diophantine graphs on n vertices, D_n, are defined, studied and generalized. The independence number, the number of vertices with full degree and the clique number of D_n are computed. Each of these quantities is the basis of a necessary condition for the existence of such a labeling.

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