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arXiv · 2412.20562

Some Necessary and Sufficient Conditions for Diophantine Graphs

M. A. Seoud·A. Elsonbaty·A. Nasr·M. Anwar
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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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M. A. Seoud, A. Elsonbaty, A. Nasr, M. Anwar. 2024-12-29. Some Necessary and Sufficient Conditions for Diophantine Graphs. https://doi.org/10.56947/gjom.v21i1.3553

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