Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
Enumeration of perfect matchings in graphs with reflective symmetry
Journal of Combinatorial Theory Series A
Graph Theory With Applications
Graph Theory With Applications
Enumeration of perfect matchings of a type of Cartesian products of graphs
Discrete Applied Mathematics
The number of spanning trees of plane graphs with reflective symmetry
Journal of Combinatorial Theory Series A
Enumeration of spanning trees of graphs with rotational symmetry
Journal of Combinatorial Theory Series A
Critical groups of graphs with reflective symmetry
Journal of Algebraic Combinatorics: An International Journal
Critical groups of graphs with dihedral actions
European Journal of Combinatorics
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As the extension of the previous work by Ciucu and the present authors [M. Ciucu, W.G. Yan, F.J. Zhang, The number of spanning trees of plane graphs with reflective symmetry, J. Combin. Theory Ser. A 112 (2005) 105-116], this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weights of spanning trees of two weighted graphs with a smaller size determined by the involution of G. As applications, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Mobius ladder, and the almost-join of two copies of a graph.