  A complete subgraph of a given graph is called a clique. A clique Polynomial
of a graph is a generating function of the number of cliques in $G$. A real
root of the clique polynomial of a graph $G$ is called a \emph{clique root} of
$G$. \\ Hajiabolhassan and Mehrabadi showed that the clique polynomial of any
simple graph has a clique root in $[-1,0)$. As a generalization of their
result, the author of this paper showed that the class of $K_{4}$-free
connected chordal graphs has also only clique roots. \\ A given graph $G$ is
called flat if each edge of $G$ belongs to at most two triangles of $G$. In
answering the author's open question about the class of \emph{non-chordal}
graphs with the same property of having only c;ique roots, we extend the
aforementioned result to the class of $K_{4}$-free flat graphs. In particular,
we prove that the class of $K_{4}$-free flat graphs without isolated edges has
$r=-1$ as one of its clique roots. We finally present some interesting open
questions and conjectures regarding clique roots of graphs.
