  The family of planar graphs is a particularly important family and models
many real-world networks. In this paper, we propose a principled framework
based on the widely-known Apollonian packing process to generate new planar
network, i.e., Type-II Apollonian network $\mathcal{A}_{t}$. The manipulation
is different from that of the typical Apollonian network, and is proceeded in
terms of the iterative addition of triangle instead of vertex. As a
consequence, network $\mathcal{A}_{t}$ turns out to be hamiltonian and
eulerian, however, the typical Apollonian network is not. Then, we in-depth
study some fundamental structural properties on network $\mathcal{A}_{t}$, and
verify that network $\mathcal{A}_{t}$ is sparse like most real-world networks,
has scale-free feature and small-world property, and exhibits disassortative
mixing structure. Next, we design an effective algorithm for solving the
problem of how to enumerate spanning trees on network $\mathcal{A}_{t}$, and
derive the asymptotic solution of the spanning tree entropy, which suggests
that Type-II Apollonian network is more reliable to a random removal of edges
than the typical Apollonian network. Additionally, we study trapping problem on
network $\mathcal{A}_{t}$, and use average trapping time as metric to show that
Type-II Apollonian network $\mathcal{A}_{t}$ has better structure for fast
information diffusion than the typical Apollonian network.
