  In this paper, we initiate the study of endomorphisms and modular theory of
the graph C*-algebras $\O_{\theta}$of a 2-graph $\Fth$ on a single vertex. We
prove that there is a semigroup isomorphism between unital endomorphisms of
$\O_{\theta}$ and its unitary pairs with a \textit{twisted property}. We
characterize when endomorphisms preserve the fixed point algebra $\fF$ of the
gauge automorphisms and its canonical masa $\fD$. Some other properties of
endomorphisms are also investigated.
  As far as the modular theory of $\O_{\theta}$ is concerned, we show that the
algebraic *-algebra generated by the generators of $\O_{\theta}$ with the inner
product induced from a distinguished state $\omega$ is a modular Hilbert
algebra. Consequently, we obtain that the von Neumann algebra
$\pi(\O_{\theta})"$ generated by the GNS representation of $\omega$ is an AFD
factor of type III$_1$, provided $\frac{\ln m}{\ln n}\not\in\bQ$. Here $m,n$
are the numbers of generators of $\Fth$ of degree $(1,0)$ and $(0,1)$,
respectively.
  This work is a continuation of \cite{DPY1, DPY2} by Davidson-Power-Yang and
\cite{DY} by Davidson-Yang.
