Some Necessary and Sufficient Conditions for Diophantine Graphs

TL;DR

This paper investigates Diophantine graphs, which are graphs with vertex labelings satisfying gcd divisibility conditions, and explores their properties, including independence number, degree, and clique number, to understand their structure.

Contribution

It introduces and studies maximal Diophantine graphs, generalizing previous concepts, and computes key graph invariants to establish necessary conditions for such labelings.

Findings

Computed independence number of D_n

Determined number of vertices with full degree in D_n

Calculated clique number of D_n

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.