The $\beta$-polynomials of Complete Graphs are Real

Xueliang Li, Ivan Gutman, Gradimir Milovanović

A polynomial is said to be real if all its zeros are real. It has been conjectured that the $\beta$-polynomials of all graphs are real. In this paper we show that the conjecture is true for complete graphs. In fact, we obtain a more general result, namely that certain linear combinations of Hermite polynomials are real.