## 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.)

### Keywords:

matching polynomial; Hermite polynomial
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