Generalizations of the matching polynomial to the multivariate independence polynomial
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 …
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 …
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 as the independence polynomial of the line graph of , 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 .
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