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 $K_{4}$-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.