Properties of Graph Based on Divisor-Euler Functions

Divisor function  gives the residues of  which divide it. A function denoted by   counts the total possible divisors of  and  gives the list of co-prime integers to . Many graphs had been constructed over these arithmetic functions. Using  and , a well known graph named as divisor Euler function graph has been constructed. In this paper, we use divisor function and anti Euler function . We label the symbol  to count those residues of  which are not co-prime to . By using these functions, we find a new graph, called divisor anti-Euler function graph (DAEFG), denoted as . Let   be a DAEFG, where  and . The objective of this sequel is to introduce and discuss the properties of DAEFG. In this work, we discuss novel classes of proposed graph with its structure using loops, cycles, components of graph, degree of its vertices, components as complete, bipartite, planar, Hamiltonian and Eulerian graphs. Also, we find chromatic number, chromatic index and clique of these graphs.