Fibonacci number of the tadpole graph

Abstract

In 1982, Prodinger and Tichy defined the Fibonacci number of a graph G to be the number of independent sets of the graph G. They did so since the Fibonacci number of the path graph P n is the Fibonacci number F n+2 and the Fibonacci number of the cycle graph C n is the Lucas number L n. The tadpole graph T n,k is the graph created by concatenating C n and P k with an edge from any vertex of C n to a pendant of P k for integers n = 3 and k = 0. This paper establishes formulae and identities for the Fibonacci number of the tadpole graph via algebraic and combinatorial methods.