Tricyclic graphs with maximum Merrifield-Simmons index

  • Authors:

  • Zhongxun Zhu;Shuchao Li;Liansheng Tan

  • Affiliations:

  • Department of Computer Science, Central China Normal University, Wuhan 430079, PR China and Department of Computer Science, South Central University for Nationalities, Wuhan 430074, PR China;Faculty of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China;Department of Computer Science, Central China Normal University, Wuhan 430079, PR China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

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)-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 n vertices having maximum Merrifield-Simmons index is determined.