  There are totally 11 kinds of Archimedean tiling for the plane. Applying the
Floquet-Bloch theory, we derive the dispersion relations of the periodic
quantum graphs associated with a number of Archimedean tiling, namely the
triangular tiling {$(3^6)$}, the elongated triangular tiling {$(3^3,4^2)$}, the
trihexagonal tiling {$(3,6,3,6)$} and the truncated square tiling {$(4,8^2)$}.
The derivation makes use of characteristic functions, with the help of the
symbolic software Mathematica.
  The resulting dispersion relations are surprisingly simple and symmetric.
They show that in each case the spectrum is composed of point spectrum and an
absolutely continuous spectrum. We further analyzed on the structure of the
absolutely continuous spectra. Our work is motivated by the studies on the
periodic quantum graphs associated with hexagonal tiling in \cite{KP} and
\cite{KL}.
