Sums of Polynomials and Clique Roots

TL;DR

This paper introduces an interlacing lemma related to sums of polynomials with real roots, characterizes graphs with only clique roots, and discusses open problems in the context of the Kadison-Singer conjecture.

Contribution

It presents a new interlacing lemma and applies it to characterize graphs with clique roots, advancing understanding in polynomial root analysis and graph theory.

Findings

The interlacing lemma provides a necessary condition for sums of polynomials to have real roots.

Characterization of certain graphs that have only clique roots.

Open problems and conjectures related to polynomial roots and graph properties.

Abstract

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.