<jats:p>Let <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
<mi>G</mi>
<mo>=</mo>
<msub>
<mrow>
<mi>G</mi>
</mrow>
<mrow>
<mn>1</mn>
</mrow>
</msub>
<mo>×</mo>
<msub>
<mrow>
<mi>G</mi>
</mrow>
<mrow>
<mn>2</mn>
</mrow>
</msub>
<mo>×</mo>
<mo>⋯</mo>
<mo>×</mo>
<msub>
<mrow>
<mi>G</mi>
</mrow>
<mrow>
<mi>m</mi>
</mrow>
</msub>
</math>
</jats:inline-formula> be the strong product of simple, finite connected graphs, and let <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
<mi>ϕ</mi>
<mo>:</mo>
<mi>ℕ</mi>
<mo>⟶</mo>
<mfenced open="(" close=")" separators="|">
<mrow>
<mn>0</mn>
<mo>,</mo>
<mi>∞</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula> be an increasing function. We consider the action of generalized maximal operator <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
<msubsup>
<mi>M</mi>
<mi mathvariant="bold">G</mi>
<mi>ϕ</mi>
</msubsup>
</math>
</jats:inline-formula> on <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
<msup>
<mrow>
<mi>ℓ</mi>
</mrow>
<mrow>
<mi>p</mi>
</mrow>
</msup>
</math>
</jats:inline-formula> spaces. We determine the exact value of <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
<msup>
<mrow>
<mi>ℓ</mi>
</mrow>
<mrow>
<mi>p</mi>
</mrow>
</msup>
</math>
</jats:inline-formula>-quasi-norm of <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
<msubsup>
<mi>M</mi>
<mi>G</mi>
<mi>ϕ</mi>
</msubsup>
</math>
</jats:inline-formula> for the case when <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
<mi>G</mi>
</math>
</jats:inline-formula> is strong product of complete graphs, where <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
<mn>0</mn>
<mo><</mo>
<mi>p</mi>
<mo>≤</mo>
<mn>1</mn>
</math>
</jats:inline-formula>. However, lower and upper bounds of <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
<msup>
<mrow>
<mi>ℓ</mi>
</mrow>
<mrow>
<mi>p</mi>
</mrow>
</msup>
</math>
</jats:inline-formula>-norm have been determined when <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
<mn>1</mn>
<mo><</mo>
<mi>p</mi>
<mo><</mo>
<mi>∞</mi>
</math>
</jats:inline-formula>. Finally, we computed the lower and upper bounds of <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
<msub>
<mrow>
<mfenced open="‖" close="‖" separators="|">
<mrow>
<msubsup>
<mi>M</mi>
<mi>G</mi>
<mi>ϕ</mi>
</msubsup>
</mrow>
</mfenced>
</mrow>
<mrow>
<mi>p</mi>
</mrow>
</msub>
</math>
</jats:inline-formula> when <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
<mi>G</mi>
</math>
</jats:inline-formula> is strong product of arbitrary graphs, where <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
<mn>0</mn>
<mo><</mo>
<mi>p</mi>
<mo>≤</mo>
<mn>1</mn>
</math>
</jats:inline-formula>.</jats:p>
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