  To adopt a practical method to calculate the action of geometrical operators
on quantum states is a crucial task in loop quantum gravity. In the series of
papers, we will introduce a graphical method, developed by Yutsis and Brink, to
loop quantum gravity along the line of previous works. The graphical method
provides a very powerful technique for simplifying complicated calculations. In
this first paper, the closed formula of volume operator is derived via the
graphical method. By employing suitable and non-ambiguous graphs to represent
the acting of operators as well as the spin network states, we use the simple
rules of transforming graphs to yield the resulting formula. Comparing with the
complicated algebraic derivation in some literatures, our procedure is more
concise, intuitive and visual. The resulting matrix elements of volume operator
is compact and uniform, fitting for both gauge-invariant and gauge-variant spin
network states.

General Relativity and Quantum Cosmology