On graphs with the third largest number of maximal independent sets

Let G be a simple and undirected graph. By mi(G) we denote the number of maximal independent sets in G. Erdős and Moser posed the problem to determine the maximum cardinality of mi(G) among all graphs of order n and to characterize the corresponding extremal graphs attaining this maximum cardinality. The above problem has been solved by Moon and Moser in [J.W. Moon, L. Moser, On cliques in graphs, Israel J. Math. 3 (1965) 23–28]. More recently, Jin and Li [Z. Jin, X. Li, Graphs with the second largest number of maximal independent sets, Discrete Mathematics 308 (2008) 5864–5870] investigated the second largest cardinality of mi(G) among all graphs of order n and characterized the extremal graph attaining this value of mi(G). In this paper, we shall determine the third largest cardinality of mi(G) among all graphs G of order n. Additionally, graphs achieving this value are also determined.