Generalized the Liouville’s and Möbius functions of graph

Let G = ( V , E ) be a simple connected undirected graph. In this paper, we define generalized the Liouville’s and Möbius functions of a graph G which are the sum of Liouville λ and Möbius μ functions of the degree of the vertices of a graph denoted by Λ ( G ) = ∑ v ∈ V ( G ) λ ( d e g ( v ) ) and M ( G ) = ∑ v ∈ V ( G ) μ ( d e g ( v ) ) , respectively. We also determine the Liouville’s and Möbius functions of some standard graphs as well as determining the relationships between the two functions with their proofs. The sum of generalized the Liouville and Möbius functions extending over the divisor d of degree of vertices of graphs is also given.