Figure 1

(Color online) Triangular zigzag graphene structure with $N=3$ atoms on the one edge. Under each $A$-type atom [indicated by red (light gray) circles] are corresponding coefficients. Dash lines and open circles indicate auxiliary $A$-type atoms in the three corners, which will help to introduce three boundary conditions. For zero-energy states all coefficients can be expressed as superpositions of coefficients corresponding to atoms from the one edge (upper row of atoms in our case).Reuse & Permissions

Figure 2

(Color online) Triangular zigzag graphene structure from Fig. 1. Above each $B$-type atom [indicated by blue (dark gray) circles] are corresponding coefficients. For convenience, we only left coefficients corresponding to auxiliary $B$-type atoms on the right edge. For zero-energy states coefficient from upper left corner $\left(c_{0,0}\right)$ determine all other coefficients in the structure. Introducing three boundary conditions from auxiliary atoms we obtain only trivial solution; zero-energy states consist of only $A$-type atoms.Reuse & Permissions

Figure 3

(Color online) Single-particle spectrum from tight-binding calculations for (a) 46 atoms triangle and (b) 38 atoms trapezoid (with $N_{row}=4$). There are four zero-energy states in the triangle (number of atoms on the one edge $N=5$) and two zero-energy states in the trapezoid in agreement with our analysis. Changing number of rows in the trapezoid we can control number of zero-energy states.Reuse & Permissions

Figure 4

(Color online) Two trapezoids (left) and bowtie structure (right). Each of two trapezoids has one zero-energy state, consisting of only $A$-type atoms [indicated by red (light gray) circles] for the upper trapezoid and consisting of only $B$-type atoms [indicated by blue (dark gray) circles] for the lower trapezoid. Connecting these two systems does not affect the zero-energy solutions since coefficients belonging to connecting atoms are zeros (the four nearest atoms to the dash line). The bowtie structure on the right has two zero-energy states: one which completely lies in upper part and consists of $A$-type atoms and second one lies in lower part and consists of $B$-type atoms.Reuse & Permissions