A clique polynomial of a finite simple graph $G=(V, E)$ is a polynomial $C(G,x)$ defined, as

\begin{equation}C(G,x)= 1 + \sum_{\emptyset \neq U \subseteq V(G): \text{G[U] is a clique}}x^{\vert U \vert},\nonumber \end{equation}

where the sum runs over all induced subgraphs $G[U]$ that are cliques (complete subgraphs) in $G$.

Hajiabolhasan and Mehrabadi proved that all roots of the clique polynomial of a triangle - free graph are real. Consequently, they presented a new elementary and algebraic proof of Mantel's theorem for triangle - free graphs. Moreover, Teimoori showed that the clique polynomial of the class of $K_{4}$ - free connected planar chordal graphs has only real roots.

Here, while reviewing the algebraic properties of the clique polynomials, we give a generalization of the previous results by presenting a new algebro - geometric proof of Turan's graph theorem using some ideas from cylindrical constructions on finite graphs. We conclude our talk, by proposing some open questions regarding clique polynomials and Turan-type results.