Let G = (V, E) be a connected graph. H denotes a family of pairwise disjoint graphs {H-v}(v is an element of V). The Zykov sum of G and H, denoted by G[H], is the graph obtained from G by replacing every vertex v of G with graph H-v and all vertices of H-u, H-v are adjacent if uv is an element of E. In this paper, we first give a decomposition formula for the independence polynomial I (G[H]; x). Then, we derive a formula expressing the Fibonacci number of G[H] in terms of the independence polynomial of graph G and the Fibonacci number of H-v. Finally, as applications, we compute the independence polynomials and the Fibonacci numbers of several interesting graphs, such as the windmill graphs, the path network and the ring network. (C) 2021 Elsevier B.V. All rights reserved.
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