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Exact Potts/Tutte polynomials for polygon chain graphs. (English) Zbl 1220.82028
In [R. Shrock and S.-H. Tsai, J. Phys. A, Math. Gen. 32, No.27, 5053–5070 (1999; Zbl 0938.82023)], exact results and connection has been established between the partition function of the zero-temperature q-state Potts antiferromagnet and the chromatic polynomial on an open cyclic graph chain. This work aims at generalizations of previous results to the finite temperature and ferromagnetic cases. Exact calculations of Potts model partition functions and the equivalent Tutte polynomials for polygon chain graphs, with open and cyclic boundary conditions, are presented. As special cases various graph-theoretic quantities are derived: flow and reliability polynomials, characteristics of the bond percolation, the number of spanning trees, of spanning forests, of connected spanning subgraphs etc.

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
68R10 Graph theory (including graph drawing) in computer science
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C40 Connectivity
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