degree in $G$ that induce a $C_5$. Let $X$ be the vertex set of such an independent set or clique in $G$ of cardinality $3$, if it exists; otherwise let $X$ be the vertex set of such an induced $5$-cycle in $G$. Then, $G[X]$ is $s$-regular with $s = 2$ or $s =
0$. Since $G$ is connected and the vertices in $X$ have all the same degree in $G$, they also must have the same number of neighbors in $V
\setminus X$ (which is non-empty since $n \ge 6$), implying that ${{\rm fd}}(G) \le n - 3$ or ${{\rm fd}}(G) \le n - 5$ in case case $G[X] =
C_5$. In all cases, we have ${{\rm fd}}(G) < n - 2$, a contradiction. Therefore, ${{\rm rep}}(G) \le 4$. [$\Box$]{}

We remark that the restriction on the order $n \ge 6$ in the statement of Proposition \[prop5\] is necessary since $G = C_5$ has order $n = 5$ and satisfies ${{\rm rep}}(G) = 5$ and ${{\rm fd}}(G) = 3 = n-2$. Both Proposition \[prop4\] and Proposition \[prop5\] give further evidence that a connected graph of large order achieving the upper bound in Theorem \[thm\_n2\] is highly non-regular. Further, Proposition \[prop4\] 