Hermite polynomials and a duality relation for matchings polynomials

Abstract

LetG be a graph onn vertices. Ak-matching inG is a set ofk independent edges. If 2k=n then ak-matching is called perfect. The number ofk-matchings inG isp(G, k). (We setp(G, 0)=1). The matchings polynomial ofG is

$$\alpha (G,x) = \sum\limits_{k = 0}^{[n/2]} {( - 1)^k p(G,k)x^{n - 2k} } $$

Our main result is that the number of perfect matchings in the complement ofG is equal to

$$(2\pi )^{ - 1/2} \int\limits_{ - \infty }^\infty {\alpha (G,x)} \exp ( - x^2 /2)dx.$$
((1))

LetK m be the complete graph onm vertices. Then α(K m ,x) is the Hermite polynomial He n (x) of degreen. Using (1) we show, amongst other results, that

$$\alpha (\bar G,x) = \sum\limits_{k = 0}^{[n/2]} {p(G,k)} \alpha (K_{n - 2k} ,x).$$

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References

  1. [1]

    R. Azor, J. Gillis andJ. D. Victor, Combinatorial applications of Hermite polynomials, manuscript.

  2. [2]

    A. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi,Higher Transcendental Functions (Bateman manuscript project), McGraw-Hill, 1953.

  3. [3]

    C. D. Godsil andI. Gutman, On the theory of the matching polynomial,J. Graph Theory,5 (1981), 137–144.

    MATH  MathSciNet  Google Scholar 

  4. [4]

    O. J. Heilmann andE. H. Lieb, Theory of monomer-dimer systems,Comm. Math. Physics,25 (1972), 190–232.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    S. A. Joni andG-C. Rota, A vector space analog of permutations with restricted position,J. Combinatorial Theory, Series A,29 (1980), 59–73.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    L. Lovász,Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  7. [7]

    J. Riordan,An introduction to Combinatorial Analysis, Wiley, 1958.

  8. [8]

    T. Zaslavsky, Complementary matching vectors and the uniform matching extension property,Europ. J. Comb. 2 (1981), 91–103.

    MATH  MathSciNet  Google Scholar 

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Godsil, C.D. Hermite polynomials and a duality relation for matchings polynomials. Combinatorica 1, 257–262 (1981). https://doi.org/10.1007/BF02579331

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AMS subject classification (1980)

  • 05 A 15
  • 05 C 99, 33 A 99