A (k, 6)-cage (k = 2, 3, 4, 5) is a 2-connected cubic plane graph that has only k-gons and hexagons as its faces. Continuing their work on (3, 6)-cages (2009) the authors investigate the combinatorial (topological and algebraic) structure of (2, 6)-cages and explicitly determine their eigenvalues and eigenvectors.