  We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of
massless Dirac fermion in two-dimensional $\alpha$-${T}_3$ lattice. By tuning
the hopping parameter $\alpha\in[0,1]$, the $\alpha$-${T}_3$ lattice
interpolates between pseudospin $S=1/2$ (graphene) and $S=1$ (${T}_3$ or dice
lattice), for $\alpha=0$ and $1$, respectively, which is followed by continuous
change of the Berry phase from $\pi$ to $0$. The range of existence for edge
states in the momentum space is determined by solving tight-binding equations
at the boundaries of the ZRs. We find that the transitions of in-gap bands from
bulk to edge states in the momentum space do not only occur at the positions of
the Dirac cones but also at additional points depending on $\alpha\notin
\{0,1\}$. The $\alpha$-${T}_3$ ZRs are mapped into stub Su-Schrieffer-Heeger
chains by performing unitary transforms of the bulk Hamiltonian. The
non-trivial topology of the bulk bands is revealed by the Majorana
representation of the eigenstates, where the $\mathbb{Z}_2$ topological
invariant is manifested by the winding numbers on the complex plane and the
Bloch sphere.
