Hand written solution required

Exercise 4. Prove that a: l . 0 0 711 :1: 1 . 0 0 :1: 0 U Hedi\"): . 0 0 0 a: l U 0 O 1 a: Hint: Show that this family of determinants satises (2). We now return to graph theory, because it has much more to tell us. Let 5 denote the complement of the graph G, which has all the same vertices as C but none of the same edges: if u and v are two vertices with no edge between them in G, then there is an edge between them in a, and if there is an edge between them in G then there isn't in 6. Also let pm(G) denote the number of perfect metehings in G. The big theorem on the matchings polynomial was proved by Chris Godsil around 1975. Theorem 1 (Godsil's Theorem). m \"2 me'PKR 2:) d1: 71' pm{G')=