We describe how the 5-regular simple planar graphs can all be obtained from an elementary family of starting graphs by repeatedly applying a few local expansion operations. The proof uses an amalgam of theory and computation. By incorporating the recursion into the canonical construction path method of isomorph rejection, a generator of non-isomorphic embedded 5-regular planar graphs is obtained with time complexity O(n 2 ) per isomorphism class. A similar result is obtained for simple planar pentangulations with minimum degree 2.

A neutral network is a subgraph of a Hamming graph, and its principal eigenvalue determines its robustness: the ability of a population evolving on it to withstand errors. Here we consider the most robust small neutral networks: the graphs that interpolate pointwise between hypercube graphs of consecutive dimension (the point, line, line and point in the square, square, square and point in the cube, and so on). We prove that the principal eigenvalue of the adjacency matrix of these graphs is bounded by the logarithm of the number of vertices, and we conjecture an analogous result for Hamming graphs of alphabet size greater than two.Keywords: 05C50, graph eigenvalue, hypercube, neutral networks, evolvabilityThe eigenvalues of neutral networks-subgraphs of Hamming graphs-is a fascinating subject, yet one which seems to have received little attention from the mathematics community. A recent surge of scientific interest has been motivated by advances in the theory of neutral evolution [1,2], in which the evolution of a mutating population is captured by spectral properties of its underlying neutral network [3].A genome is the set of all genotypes, or a-ary strings, of length d and alphabet size a. Typically a is small: a = 2 (hydrophilic and hydrophobic), a = 4 (nucleic acids) or a = 20 (amino acids). On the other hand, d can range from 3 (codons) to 10 8 (chromosomes). We represent the genome by a d-dimensional, where K a is the complete graph on a vertices and is the Cartesian product [4]. Each of the a d vertices in the Hamming graph corresponds to a genotype, and two vertices share an edge if the genotypes differ by a single mutation (Hamming distance one). A neutral network is the set of genotypes with the same phenotype (observable characteristics); it is just a subgraph of H d,a . In this Note we use neutral network and phenotype interchangably.Assuming the phenotype has achieved high fitness, adjoining phenotypes will have a relatively negligible growth rate and act as effective absorbing boundaries. Now consider a mutating population on the neutral network. That portion which mutates off of it will be lost, whereas that portion which stays on will survive. The robustness r of a neutral network is the long-term probability that a randomly selected individual mutating in a random direction survives. It is the principal eigenvalue λ of the adjacency matrix of the neutral network divided by the number of directions for mutation: r = λ/(d (a − 1)). This Note is about the maximum robustness that a neutral network can have, which for small neutral networks are themselves Hamming graphs and interpolations between them, shown for a = 2 in Figure 1. Surprisingly, this ceases to be true for larger neutral networks on longer sequences (d = 19 and above),

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