Tricyclic graphs with maximum Merrifield–Simmons index

It is well known that the graph invariant, ‘the Merrifield–Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield–Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16–21]. The sharp upper and lower bounds for the Merrifield–Simmons indices of unicyclic graphs are characterized by Pedersen and Vestergaard [A.S. Pedersen, P.D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005) 246–256]. The sharp upper bound for the Merrifield–Simmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The Merrifield–Simmons index in (n,n+1)(n,n+1)-graphs, J. Math. Chem. 43 (1) (2008) 75–91]. The sharp lower bound for the Merrifield–Simmons index of bicyclic graphs is determined by Jing and Li [W. Jing, S. Li, The number of independent sets in bicyclic graphs, Ars Combin, 2008 (in press)]. In this paper, we will consider the tricyclic graph, i.e., a connected graph with cyclomatic number 3. The tricyclic graph with nn vertices having maximum Merrifield–Simmons index is determined.