Diophantine Graphs

TL;DR

This paper defines linear Diophantine labeling, a bijection $f$ from the vertex set to $\{1,\dots,n\}$ such that for each edge $uv$, $\gcd(f(u),f(v))$ divides $n$, tying Diophantine graphs to prime graphs. It derives an explicit edge-count formula for maximal Diophantine graphs $D_n$ via inclusion-exclusion over prime divisors of $n$, and provides a precise degree formula for any vertex using the reduced label $f^*(u)=\frac{f(u)}{(f(u),n)}$ and $p$-adic valuations $v_p(n)$. The paper identifies full-degree labels as those with $f(u)=p^{\acute{v}_p(n)}$ in the specified range or $f(u)\mid n$, and proves several necessary and sufficient conditions for equality of degrees, including divisibility-based relations between labels. It also establishes that infinitely many Diophantine graphs are not prime graphs and discusses potential algorithmic approaches and applications at the intersection of graph theory and number theory.

Abstract

This manuscript introduces Diophantine labeling, a new way of labeling of the vertices for finite simple undirected graphs with some divisibility condition on the edges. Maximal graphs admitting Diophantine labeling are investigated and their number of edges are computed. Some number-theoretic techniques are used to characterize vertices of maximum degree and nonadjacent vertices. Some necessary and sufficient conditions for vertices of equal degrees are found.

Figures (2)

• Figure 1: $P$ is a Diophantine graph while $K_5$ is not
• Figure 2: Maximal Diophantine Graphs $D_4,\dots, D_{8}$

Theorems & Definitions (52)

• Theorem 1.1
• Definition 1.1
• Definition 2.1
• Definition 2.2
• Example 2.1
• Example 2.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof