  Carbon allotropes have a large family of materials with varieties of crystal
structures and properties and can realize different topological phases. Using
first principles calculations, we predict a new two-dimensional (2D) carbon
allotrope, namely penta-octa-graphene, which consists of pentagonal and
octagonal carbon rings. We find that penta-octa-graphene can host both type-I
and type-II Dirac line nodes (DLNs). The band inversion between conduction and
valence bands forms the type-I DLNs and the two highest valence bands form the
type-II DLNs. We find that the type-I DLNs are robust to the biaxial strain and
the type-II DLNs can be driven to type-I when applying over 3 $\%$ biaxial
stretching strain. A lattice model based on the $\pi$ orbitals of carbons is
derived to understand the coexistence mechanism of type-I and type-II DLNs in
penta-octa-graphene. Possible realizations and characterizations of this
penta-octa-graphene in the experiment are also discussed. Our findings shed new
light on the study of the coexistence of multiple topological states in the 2D
carbon allotropes.
