STRUCTURES AND METRIC DIMENSIONS OF DIVISOR EULER FUNCTION GRAPH STRUCTURES AND METRIC DIMENSIONS OF DIVISOR EULER FUNCTION GRAPH G(D(\phi(t))

The symbol phi(t) counts positive integers less than that are coprime to , whereas represents the number of divisors of . 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 , where the edge set is based on . The metric representation of any arbitrary vertex with respect to an ordered subset of , is the -vector, written as, such that is the shortest distance between the vertices 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.