Spectral properties of the He matrix of hexagonal systems

The He matrix, put forward by He and He in 1989 [31], is designed as a means for uniquely representing the structure of a hexagonal system (= benzenoid graph). Observing that the He matrix is just the adjacency matrix of a pertinently weighted inner dual of the respective hexagonal system, we establish a number of its spectral properties. The spectral radius of the He matrix is less than 12, but can be arbitrarily close to 12. In case of catacondensed systems, the spectral radius is less than 6. Based on a computer search, we conjecture that the naphthalene graph is the only hexagonal system whose He matrix has integral spectrum. Some results for the energy of the He matrix are also obtained.