We present a new heuristic method for minimising crossings in a graph. The method is based upon repeatedly solving the so-called star insertion problem in the setting where the combinatorial embedding is fixed, and has several desirable characteristics for practical use. We introduce the method, discuss some aspects of algorithm design for our implementation, and provide some experimental results. The results indicate that our method compares well to existing methods, and also that it is suitable for dense instances.

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A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction generates a large proportion of the cubic graphs, which are cospectral with another cubic graph.

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