The number of independent sets in bipartite graphs and benzenoids

Given a graph G, we study the number of independent sets in G, denoted i(G). This parameter is known as both the Merrifield–Simmons index of a graph as well as the Fibonacci number of a graph. In this paper, we give general bounds for i(G) when G is bipartite and we give its exact value when G is a balanced caterpillar. We improve upon a known upper bound for i(T) when T is a tree, and study a conjecture that all but finitely many positive integers represent i(T) for some tree T. We also give a recursive formula for finding i(G) when G is a linear chain of hexagons and use this to study the number of independent sets in benzenoids. We also answer a conjecture relating i(G) when G is a linear chain of hexagons and the number of \(2\times n\) matrices containing a 1 in the top left entry where all entries are integer values and adjacent entries differ by at most 1.