Tricyclic graphs with maximum MerrifieldSimmons index ?
It is well known that the graph invariant, the MerrifieldSimmons index is important one in structural chemistry. The connected acyclic graphs with maximal and minimal MerrifieldSimmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 1621]. The sharp upper and lower bounds for the MerrifieldSimmons 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) 246256]. The sharp upper bound for the MerrifieldSimmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The MerrifieldSimmons index in (n,n+1) ( n , n + 1 ) -graphs, J. Math. Chem. 43 (1) (2008) 7591]. The sharp lower bound for the MerrifieldSimmons 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 n n vertices having maximum MerrifieldSimmons index is determined.
tricyclic graph merrifieldsimmons index independent set extremal graph