SOME IDENTITIES ARISING FROM THE FIBONACCI NUMBERS OF CERTAIN GRAPHS GLENN HOPKINS and WILLIAM STATON University of Mississippi, University, MS 38677 (Submitted December 1982) Tichy and Prodinger [5] have defined the Fibonacci number of a graph G to be the number of independent vertex sets J in G; recall that I is independent if no two of its vertices are adjacent. Following Tichy and Prodinger, we de- note the Fibonacci number of G by F(G). If k is a nonnegative integer, we will denote the /c-element independent vertex sets in G by F k (G) . It is clear that E F k (G) = F(G). Kreweras [4] (see also [3]) has introduced the notion of the Fibonacci polynomial, k>o y K j We define the more general concept of the Fibonacci polynomial of a graph G 9 denoted F G (x). In case G is a path on n vertices, which closely resembles Kreweras 1 polynomial. Before defining F G (x), we compute F k (P n ) , P n the path on n vertices, and F k (C n ) , C n the cycle on n vertices. Proposition 1 (i) V p *> = !; (ii) F 1 (P n ) = n; (iii) F k (P n + 0 = F k (PJ + ^-i(P»-i> for 1 < fe < [^p (iv) M PJ-C 1 -^ 1 ) for 0 < * < [2-ti]. Proof: The first two statements are obvious. To verify (iii), consider those ^-element independent sets that contain the initial point of the path and those that do not. Finally, (iv) may be verified using (iii) and induction on n. • Proposition 1 provides a natural graph-theoretic interpretation of the well-known formula T.( n ' k k + l )=F n+1 , k>0 X K ' the n + 1 th Fibonacci number. The right side of the equality is the number of independent sets of a path with n vertices. The left side is the sum over all k of the number of k-element independent sets. The following proposition will enable us to give an analogous identity involving Lucas numbers, and a graph- theoretic interpretation of that identity. • 1984] 255