The symbol  counts positive integers less than t that are co-prime to t, whereas  represents the number of divisors of t. We construct a new number theoretic graph called Divisor Euler Function Graph (DEFG), labeled as  by incorporating  and  The motivation for this work was to discover the structures of these newly defined graphs (DEFG), followed by an attempt to discover the metric dimensions of such graphs. In  the vertex set  assumes divisors of t, where the edge set  is based on  The metric representation of any arbitrary vertex v with respect to an ordered subset  of  is the k-vector, written as,  such that  is the shortest distance between the vertices v and  In this piece of work, we discuss structures of DEFG such as loops, cycles and their lengths, connectedness, maximal and minimal degree of vertices, components of complete graphs as bipartite graphs, planarity, Hamiltonicity, and Eulerianity. We further find chromatic number, chromatic index and clique numbers of these graphs. Also, we compute distance codes, resolving sets, local resolving sets, metric dimension and local metric dimension of certain families in DEFG.

graph of divisor function  Euler function graph  divisor Euler function graph  divisor Euler function sub-graph  resolvent  metric dimension.

Received: October 8, 2024; Revised: November 22, 2024; Accepted: December 2, 2024; Published: March 8, 2025

How to cite this article: Asif Abd ur Rehman and M. Khalid Mahmood, Structures and metric dimensions of divisor Euler function graph  Advances and Applications in Discrete Mathematics 42(4) (2025), 343-361. https://doi.org/10.17654/0974165825022

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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