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arXiv · 1610.00805

Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

Jonathan Leake·Nick Ryder
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Abstract

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph of $G$, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil's theorems on the divisibility of matching polynomials of trees related to $G$.

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Jonathan Leake, Nick Ryder. 2019-02-19. Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial. https://arxiv.org/abs/1610.00805

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