Solution
= Solution
For lattice spacing $2a$, the first Brillouin zone is
$$
-\frac{\pi}{2a}\leq k\leq\frac{\pi}{2a}.
$$
At its boundaries $E_\pm=\pm t$, while at $k=0$ the energies are $E_\pm=\pm\sqrt5,t$. Thus the lower band runs from $-\sqrt5,t$ to $-t$, the upper from $t$ to $\sqrt5,t$, and each has bandwidth
$$
(\sqrt5-1)t.
$$
The requested plot consists of these two even branches, separated by a gap of size $2t$ at the zone boundaries.