For any $n \times n$ nonnegative matrix $A$, and any norm $\|.\|$ on $\mathbb{R}^n$, $\eta_{\|.\|}(A)$ is defined as $ \sup\ \{\frac{\|A \otimes x\|}{\|x\|} :\ x\in \mathbb{R}_+^n \ , \ x\neq 0\}.$ Let $P(\lambda)$ be a matrix polynomial in the max algebra. In this paper, we introduce $\eta_{\|.\|}[P(\lambda)]$, as a generalization of the matrix norm $\eta_{\|.\|}(.)$, and we investigate some algebraic properties of this notion. We also study some properties of the maximum circuit geometric mean of the companion matrix of $P(\lambda)$ and the relationship between this concept and the matrices $P(1)$ and coefficients of $P(\lambda)$. Some properties of $\eta_{\|.\|}(\Psi)$, for a bounded set of max matrix polynomials $\Psi$, are also investigated.
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