  In this paper, pursuing the same line of ideas in the proof of an old
longstanding open conjecture of \emph{Kadison-Singer} , we introduce a key
lemma which we call it the interlacing lemma which indicates a necessary
condition for having a real root for sums of polynomials with (at least) one
real root. Then, as an immediate application of this simple but potentially
useful lemma we characterize several class of graphs which have only clique
roots. Finally, we conclude our paper with several interesting open problems
and conjectures for interested readers.
