Maximum Independent Set of Cliques and The Generalized Mantel's Theorem

A complete subgraph of any simple graph $G$ on $k$ vertices is called a $k$-\emph{clique} of $G$. In this paper, we first introduce the concept of the value of a $k$-clique ($k>1$) as an extension of the idea of the degree of a given vertex. Then, we obtain the generalized version of handshaking lemma which we call it clique handshaking lemma. The well-known classical result of Mantel states that the maximum number of edges in the class of triangle-free graphs with $n$ vertices is equal to $\frac{n^{2}}{4}$. Our main goal here is to find an extension of the above result for the class of $K_{\omega+1}$-free graphs, using the ideas of the value of cliques and the clique handshaking lemma.