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The \(\beta\)-polynomials of complete graphs are real. (English) Zbl 0946.05068

The matching polynomial \(\alpha (F,x)\) of a graph \(G\) has the numbers of the \(k\)-matchings of \(G\) as its coefficients. The authors prove that for any circuit \(C\) of \(G\) all zeros of the polynomial \(\beta(G,C,x) = \alpha(G,x) - 2\alpha(G\setminus C,x)\) are real if \(G\) is a complete graph. The authors use the fact that \(\alpha(K_n,x)\) is a Hermite polynomial.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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