On the third largest number of maximal independent sets of graphs∗

Given a graph G = (VG,EG), a set I ⊆VG is independent if there is no edge of G between any two vertices of I. A maximal independent set is an independent set that is not a proper subset of any other independent set. The dual of an independent set is a clique, in the sense that clique corresponds to an independent set in the complement graph. The set of all maximal independent sets of a graph G is denoted by MI(G) and its cardinality by mi(G). Given a simple graph G = (VG,EG), the cardinality of VG is called the order of G. G−v denotes the graph obtained from G by deleting vertex v ∈VG (this notation is naturally extended if more than one vertex is deleted). For v ∈VG, let NG(v) (or N(v) for short) denote the set of all the adjacent vertices of v in G and d(v) = |NG(v)|, the degree of v in G. In particular, let ∆(G) = max{d(x)|x ∈ VG} and δ (G) = min{d(x)|x ∈ VG}. For convenience, let NG[x] = {x}∪NG(x). A leaf of G is a vertex of degree one. For any two graphs G and H, let G⊎H denote the disjoint union of G and H, and for any nonnegative integer t, let tG stand for the disjoint union of t copies of G. For a connected graph H with maximum degree vertex x and a graph G = G1 ⊎G2 ⊎·· ·⊎Gk with ui being the maximum degree vertex in Gi, i = 1,2, . . . ,k, define the graph H ∗G to be the graph with vertex set VH∗G =VH ∪VG and edge set EH∗G = EH ∪EG ∪{xui : i = 1,2, . . . ,k}. Throughout the text we denote by Pn,Cn,Kn and K1,n−1 the path, cycle, complete graph and star on n vertices, respectively. Further on we need the following lemmas.