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.
