Eigenvalues of neutral networks: Interpolating between hypercubes

The first 16 ‘‘bricklayer’s graphs’’ and the principal eigenvalue of their adjacency matrices.

Discrete Mathematics 339, 1283 (2016)

T. Reeves, R. Farr, J. Blundell, A. Gallagher, T. Fink

The first 16 ‘‘bricklayer’s graphs’’ and the principal eigenvalue of their adjacency matrices.

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.