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Coefficients of the Matching Polynomial of a Self-Complementary Graph

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Abstract

Let \(G=(V,E)\) be a simple graph and \(\overline{G}\) be its complement. It is well-known that the matching polynomial of G is completely determined by that of \(\overline{G}\). We are curious about what one can deduce if G is a self-complementary graph. Suppose that G is a self-complementary graph with \(n=4t\) or \(4t+1 (t\ge 1)\) vertices, and its matching polynomial is \(\mu (G,x)=\sum _{r=0}^{2t}(-1)^{r}p(G,r)x^{n-2r}\). In this paper, we first deduce that the coefficients of \(\mu (G,x)\) satisfy the recurrent relation

$$\begin{aligned} p(G,2r+1)=\frac{1}{2}\sum _{i=0}^{2r}(-1)^{i}p(G,i)p(K_{n-2i},2r-i+1), \ \ 0\le r\le t-1, \end{aligned}$$

where \(K_n\) denotes the complete graph of order n. Then we show that, in addition to p(G, 2), p(G, 3) is also completely determined by the degree sequence of G, and the explicit expressions in terms of its degree sequence are given. Finally, p(G, 2) and p(G, 3) are computed for all self-complementary graphs with \(n\le 13\) vertices.

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References

  1. Broere, I., Hattingh, J. H.: On the construction of self-complementary circulant graphs, in Graph theory, combinatorics, and algorithms, Vol. 1, 2 (Kalamazoo, MI, 1992), Wiley, New York pp. 123-129 (1995).

  2. Cao, Y., Chen, H., Wang, S.: Self-complementary (Pseudo-)Split Graphs, In: Soto, J.A., Wiese, A.: (eds) LATIN 2024: Theoretical Informatics. Lecture Notes in Computer Science , Vol 14579. Springer, Cham, (2024)

  3. Carrillo, L.D.: Hamiltonian-connected self-complementary graphs, in Graph theory and applications (Hakone, 1990). Discrete Math. 127, 75–93 (1994)

    Article  MathSciNet  Google Scholar 

  4. Clapham, C.R.J.: Potentially self-complementary degree sequences. J. Combin. Theory 20(B), 75–79 (1976)

    Article  MathSciNet  Google Scholar 

  5. Clapham, C.R.J.: An easier enumeration of self-complementary graphs. Proc. Edinburgh Math. Soc. (2) 27, 181–183 (1984)

    Article  MathSciNet  Google Scholar 

  6. Farrugia, A.: Self-complementary graphs and generalisations: a comprehensive reference manual, Master thesis, University of Malta, (1999)

  7. Fronček, D., Rosa, A., Širáň, J.: The existence of self-complementary circulant graphs. European J. Combin. 17, 625–628 (1996)

    Article  MathSciNet  Google Scholar 

  8. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)

    Google Scholar 

  9. Gutman, I.: The matching polynomial. MATCH Commun. Math. Comput. Chem. 6, 75–91 (1979)

    MathSciNet  Google Scholar 

  10. Hartsfield, N.: On regular self-complementary graphs. J. Graph Theory 11, 537–538 (1987)

    Article  MathSciNet  Google Scholar 

  11. Jajcay, R., Li, C.H.: Constructions of self-complementary circulants with no multiplicative isomorphisms. European J. Combin. 22, 1093–1100 (2001)

    Article  MathSciNet  Google Scholar 

  12. Li, C.H.: On self-complementary vertex transitive graphs, Commun. Algebra 25, 3903–3908 (1997)

    Article  MathSciNet  Google Scholar 

  13. Lovász, L., Plummer, M. D.: Matching Theory, Annals Discrete Math. North-Holland, Amsterdam 29 (1986).

  14. Manna, P., Mehatari, R.: On the self-complementary power graph of finite groups. Indian J. Pure Appl. Math. (2025). https://doi.org/10.1007/s13226-025-00751-3

    Article  Google Scholar 

  15. Mathon, R.: On self-complementary strongly regular graphs. Discrete Math. 69, 263–281 (1988)

    Article  MathSciNet  Google Scholar 

  16. Nair, P.S.: Construction of self-complementary graphs. Discrete Math. 175, 283–287 (1997)

    Article  MathSciNet  Google Scholar 

  17. Trotignon, N.: On the structure of self-complementary graphs. Electron. Notes Discret. Math. 22, 79–82 (2005)

    Article  Google Scholar 

  18. Rao, S.B.: On regular and strongly-rugular self-complementary graphs. Discrete Math. 54, 73–82 (1985)

    Article  MathSciNet  Google Scholar 

  19. Read, R.C.: On the number of self-complementary graphs and digraphs. J. London Math. Soc. 38, 99–104 (1963)

    Article  MathSciNet  Google Scholar 

  20. Ringel, G.: Selbstkomplementäre Graphen. Arch. Math. 14, 354–358 (1963)

    Article  MathSciNet  Google Scholar 

  21. Rodger, C.A.: Self-complementary graph decompositions. J. Aust. Math. Soc. (A) 53, 17–24 (1992)

    Article  MathSciNet  Google Scholar 

  22. Rodl, V., Sinajova, E.: Note on Ramsey numbers and self-complementary graphs. Math. Slovaca 45, 243–249 (1995)

    MathSciNet  Google Scholar 

  23. Sachs, H.: ber Selbstkomplementare Graphen. Publ. Math. Debrecen 9, 279–288 (1962)

    Google Scholar 

  24. Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8, 410–421 (1979)

    Article  MathSciNet  Google Scholar 

  25. Xu, J., Wong, C.K.: Self-complementary graphs and Ramsey numbers Part-I: the decomposition and construction of self-complementary graphs. Discrete Math. 223, 309–326 (2000)

    Article  MathSciNet  Google Scholar 

  26. Xu, J.: Self-complementary Graph Theory with Applications (in chinese). Xidian University Press, Xi’an (1999)

    Google Scholar 

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Authors and Affiliations

  1. School of Sciences, Jimei University, Fujian, China

    Haiyan Chen & Yinxia Yuan

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  1. Haiyan Chen
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  2. Yinxia Yuan
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Correspondence to Yinxia Yuan.

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We declare that we have no conflict of interest.

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Communicated by Yoshio Okamoto.

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H.Y.Chen was supported by the National Natural Science Foundation of China (grant numbers 12271210, 12071180).

Appendices

Appendix A. Matlab code

code 1

check for duplicates

figure a

code 2

the number of max-matchings

figure b

code 3

Example of \(G_1\) with \(n=8\).

Number the vertices of \(G_1\) first (See Fig. 4)

Fig. 4
figure 4

\(G_1\) with \(n=8\)

Full size image
figure c

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Chen, H., Yuan, Y. Coefficients of the Matching Polynomial of a Self-Complementary Graph. Graphs and Combinatorics 42, 22 (2026). https://doi.org/10.1007/s00373-026-03021-z

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