Then the following holds.\
[(a)]{} If $n \ge 2$ and $\delta(G) \ge 1$, then ${{\rm fd}}(G) \le n - n/(({{\overline{d}}}(G) + 1) \Delta(G))$.\
[(b)]{} ${{\rm fd}}(G) \le n  - n/(2{{\overline{d}}}(G) - 2\delta(G) +1)\chi(G)$.\
[(c)]{} For $r \ge 2$, if $G$ is an $r$-regular graph, then ${{\rm fd}}(G)
\le rn/(r+1)$. \[prop2\]

**Proof.** (a) Let $B$ be a maximum independent set in $G$. Since $G$ has no isolated vertex, the set $V \setminus B$ is a dominating set in $G$. We now consider the degrees of the vertices of $B$. There are at most ${{\rm span}}(G)$ possible distinct values for these degrees, and so, by the Pigeonhole Principle, at least one value, say $q$, appears at least $\alpha(G)/{{\rm span}}(G)$ times. Let $Q$ be the set of all vertices in $B$ with degree $q$. Then, $Q \subseteq B$ and $|Q| = q$. Let $D = V \setminus Q$. Then, $D$ is a FD-set, and so ${{\rm fd}}(G) \le n
- |Q| \le n - \alpha(G)/{{\rm span}}(G)$. The desired result now follows from the observation that ${{\rm span}}(G) \le \Delta(G)$ and from the Caro-Wei Theorem.

\(b) Let ${{\rm rep}}(G) = m$ and suppose that $X =
\{v_1,\ldots,v_m\}$ is a set of vertices with the 