Graphs with the second largest number of maximal independent sets

On graphs with the third largest number of maximal independent sets

“…If d( y) = 3, we have the following results obtained by Jin and Li in [12]: Fig. 2 for graphs L i (i = 1, 2, 3)).…”

“…Also, from [12], we know that mi(G 1 ∪ ( n−1 ( n−1 3 − 1)K 3 ) = 3 4 g(n) < 97 108 g(n) = i(n). This completes the proof of this case.…”

“…Recently, Sagan and Vatter [10] settled the problem for the family of graphs with at most r cycles and its connected counterpart, and Goh et al [11] independently obtained the same result under the condition that the order of the graph is no less than 3r. More recently, Jin and Li [12] investigated the second largest cardinality of mi(G) among all graphs of order n. 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.…”

On the Third Largest Number of Maximal Independent Sets of Graphs

“…The last inequality follows from [5]. Hence, in view of the expression of i(n) in (1.1) we have mi(C n ) < i(n).…”

Trees with the second largest number of maximal independent sets

“…It was then extensively studied for various classes of graphs in the literature, including trees, forests, (connected) graphs with at most one cycle, bipartite graphs, connected graphs, k-connected graphs, (connected) triangle-free graphs; for a survey see [4]. Recently, Jin and Li [2] determined the second largest number of maximal independent sets among all graphs of order n.…”