Diophantine Graphs

TL;DR

This paper introduces Diophantine labeling for graphs, explores maximal graphs with this property, and uses number theory to analyze vertex degrees and adjacency conditions.

Contribution

It presents a novel Diophantine labeling method for graphs and characterizes maximal graphs and vertex degree conditions using number-theoretic techniques.

Findings

Maximal graphs with Diophantine labeling are characterized and their edge counts are computed.

Number-theoretic methods are used to analyze vertex degrees and adjacency.

Conditions for vertices of equal degrees are established.

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.