Multiplicity of negative one of independence polynomials of graphs

We initiate the study of the multiplicity of negative one of independence polynomials of graphs. In this article, we simply refer to this as the \emph{multiplicity} of a graph. As applications, we provide a graph-theoretic description of trees whose independence complexes are contractible, give a new sufficient condition for independence polynomials of graphs to be log-concave, and finally, determine possible pairs $(\operatorname{mult}_{-1}P_G, \alpha(G))$, where $P_G$ denotes the independence polynomial of $G$, and $\alpha(G)$ the independence number. The study of the pairs $(\operatorname{mult}_{-1}P_G, \alpha(G))$ is equivalent to finding all pairs of the numerator degree and denominator degree of the Hilbert series of the edge ideal of $G$. We also use spectral graph theory to obtain results on the multiplicity of line graphs of forests. Finally, we give some translations and applications in combinatorial commutative algebra.