The Fibonacci numbers of certain subgraphs of circulant graphs

Abstract

The Fibonacci number ℱ(G) of a graph G with vertex set V(G), is the total number of independent vertex sets S⊂V(G); recall that a set S⊂V(G) is said to be independent whenever for every two different vertices u,v∈S there is no edge between them. In general, the problem to find the Fibonacci number of a graph is NP-complete. Prodinger and Tichy proved that the Fibonacci number of Pn, the path of order n is Fn+2, the n+2-Fibonacci number; and the Fibonacci number of Cn, the cycle of order n, is Ln, the n-Lucas number.A circulant graph Cn(m1,m2,…,mr) is a graph of order n with vertex set V={v1,v2,…,vn} and edge set E={vivi+mj(modn):i∈{1,2,…,n},j∈{1,2,…,r}}, where r∈Z+. The values mj are the jump sizes. In this paper we only deal with the circulant graphs of order n with r consecutive jumps 1,2,…,r. Cn(1,2,…,r) is denoted by Cn[r].We prove that the total number of independent vertex sets of the family of graphs Cn[r] for all n≥r+1, and for several subgraphs of this family is completely determined by some sequences which are constructed recursively like the Fibonacci and Lucas sequences, even more, these new sequences generalize the Fibonacci and Lucas sequences.