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In this paper, the authors establish the bounds for the Hardy-Littlewood maximal operator defined on a finite directed graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>G</mi><mo>→</mo></mover></semantics></math></inline-formula> in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>BV</mi><mi>p</mi></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>G</mi><mo>→</mo></mover><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> of bounded <i>p</i>-variation functions. More precisely, the authors obtain the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>BV</mi><mi>p</mi></msub></semantics></math></inline-formula> norms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mover accent="true"><mi>G</mi><mo>→</mo></mover></msub></semantics></math></inline-formula> for some directed graphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>G</mi><mo>→</mo></mover></semantics></math></inline-formula>.