Abstract
Explicit recurrences are derived for the matching polynomials of the basic types of hexagonal cacti, the linear cactus and the star cactus and also for an associated graph, called the hexagonal crown. Tables of the polynomials are given for each type of graph. Explicit formulae are then obtained for the number of defect‐d matchings in the graphs, for various values of d. In particular, formulae are derived for the number of perfect matchings in all three types of graphs. Finally, results are given for the total number of matchings in the graphs.
Highlights
The graphs considered here will be finite and without loops or multiple edges.Let G be such a graph
Let us associate with each node and edge of G the respectively, and with each matching in G, the weight w and w2
13, we obtain the following theorem which gives an explicit formula for the total number of matchings in a hexagonal crown
Summary
The graphs considered here will be finite and without loops or multiple edges. Let G be such a graph. Let us associate with each node and edge of G the respectively, and with each matching in G, the weight w and w2. We will deduce explicit formulae for the number of defect-d matchings in these graphs, for various values of d. By partitioning the matchings in G according to whether or not they contain the edge e, we obtain the following result. Let G be a graph with p nodes and q edges. Consider the spanning subgraphs of G with two edges These will consist of the matchings with two edges and the spanning subgraphs with a path of length 2 and p-3 isolated nodes. Let G consist of a graph G with the chain attached to node x. The graph obtained from L by attaching two chains of length 2 to one of its terminal nodes, will be denoted by An (see Figure (ii)). The result follows by substituting for G in the equation above
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