On clique values identities and mantel-type theorems

In this paper, we first extend the weighted handshaking lemma, using a generalization of the concept of the degree of vertices to the values of graphs. This edge-version of the weighted handshaking lemma yields an immediate generalization of the Mantel's classical result which asks for the maximum number of edges in triangle-free graphs to the class of K4-free graphs. Then, by defining the concept of value for cliques (complete subgraphs) of higher orders, we also extend the classical result of Mantel for any graph G. We finally conclude our paper with a discussion about the possible future works.