On the Fibonacci numbers of the composition of graphs

Let G = (V, E) be a graph. A subset S of V is said to be independent if for every two vertices u, v is an element of S there is no edge between them. The Fibonacci number of a graph G is the total number of independent vertex sets of G. The problem of finding the Fibonacci number of a graph is an NP-complete Problem. A closely related concept is the independence polynomial of a graph, whose kth coefficient equals the number of independent sets of cardinality k in the graph. The composition of two graphs G and H is the graph obtained by replacing every vertex of G with a copy of H and all vertices of two copies are adjacent if the corresponding vertices in G are adjacent. Previous results on Fibonacci numbers of graphs, allow us to calculate the Fibonacci number of very few particular classes of graphs. In this paper, we show some new nice qualitative results, which allow us to calculate the Fibonacci numbers of infinitely many classes of graphs, obtained by the composition of graphs. In particular, we calculate the Fibonacci number of the composition of G and H in terms of the independence polynomial of G and the Fibonacci number of H. (C) 2019 Published by Elsevier B.V.