A real number $\sigma$ is a shortness exponent of a family $\Gamma$ of graphs if there exists a real number $b$ and a sequence $\{G_m\}$ of nonisomorphic graphs $G_m$ from $\Gamma$ such that $h(G_m)\leq b[n(G_m)]^\sigma$, where $h(G)$ is the length of the longest circuit of $G$ and $n(G)$ is the number of nodes of $G$. The authors find bounds for the shortness exponent of families of $d$-polyhedral graphs with $d>3$ and with bounded maximal valency. For other results on the shortness exponent see an article by B. Grünbaum and the second author [J. Combinatorial Theory Ser. A {\bf 14} (1973), 364--385; MR0314691 (47 \#3242)]. \par \{For the entire collection see MR0363962 (51 \#217).\}