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arXiv · 2607.08480

Multiplicity of negative one of independence polynomials of graphs

Om Prakash Bhardwaj·Trung Chau·Sayeed Ikram·Gargi Lather·Vasudeva Nanjangud·Chitra Venugopal
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Abstract

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.

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Om Prakash Bhardwaj, Trung Chau, Sayeed Ikram, Gargi Lather, Vasudeva Nanjangud, Chitra Venugopal. 2026-07-09. Multiplicity of negative one of independence polynomials of graphs. https://arxiv.org/abs/2607.08480

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