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

Frobenius-Perron theory of the bound quiver algebras containing loops