  The spectral radius of matrix, also known as Frobenius-Perron dimension, is a
useful tool for studying linear algebras and plays an important role in the
classification of the representation categories of algebras. In this paper, we
study the Frobenius-Perron theory of the representation categories of bound
quiver algebras containing loops, find a way to calculate the Frobenius-Perron
dimension of these algebras when they satisfy the commutativity condition of
loops. As an application, we prove that the Frobenius-Perron dimension of the
representation category of a modified ADE bounded quiver algebra is equal to
the maximum number of loops at a vertex. Finally, we point out that there also
exists infinite dimensional algebras whose Frobenius-Perron dimension is equal
to the maximal number of loops by giving an example.
