The maximum number of independent sets of given size at least 3 in Halin graphs

For a graph G with any integer t ≥ 0, we denote by σt(G) the number of independent sets of size t in G. A Halin graph is a plane graph which consists of a plane embedding of a tree T of order at least 4 without a vertex of degree 2 and a cycle C connecting all leaves of T. In this paper we prove that the wheel uniquely maximizes the number of independent sets of given size at least 3 among all Halin graphs of fixed order. Moreover, two relevant problems are proposed to the number of independent sets of given size t.