The smallest Merrifield–Simmons index of -graphs

The Merrifield–Simmons index of a graph is defined as the total number of its independent sets. In this paper, we show that the smallest Merrifield–Simmons index of ( n , n + 1 ) -graphs is f ( n ) + 2 f ( n − 2 ) + f ( n − 4 ) , where f ( n ) is the n th Fibonacci number, and characterize the extremal graphs.