Generalized Euler’s $\Phi_w$-function and the divisor sum $T_{k_w}$-function of edge weighted graphs differential equations

In this work, generalized Euler’s $\Phi_w$-function of edge weighted graphs is defined which consists of the sum of the Euler’s $\varphi$-function of the weight of edges of a graph and we denote it by $\Phi_w(G)$ and the general form of Euler’s $\Phi_w$-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 $\sigma_k$-function for the weighted of edges of a graph $G$. It is determined a relation between generalized Euler’s $\Phi_w$-function and generalized divisor sum $T_{k_w}$-function of edge weighted graphs.