Generalized Euler’s Φ w -function and the divisor sum T k w -function of edge weighted graphs differential equations

In this work, generalized Euler’s Φ w -function of edge weighted graphs is defined which consists of the sum of the Euler’s φ -function of the weight of edges of a graph and we denote it by Φ w ( G ) and the general form of Euler’s Φ w -function of some standard edge weighted graphs is determined. Also, we define the divisor sum T k w -function T k w ( G ) of the graph G , which is counting the sum of the sum of the positive divisor σ k -function for the weighted of edges of a graph G . It is determined a relation between generalized Euler’s Φ w -function and generalized divisor sum T k w -function of edge weighted graphs.