  For any graph $G$ of order $p$, a bijection $f: V(G)\to [1,p]$ is called a
numbering of the graph $G$ of order $p$. The strength $str_f(G)$ of a numbering
$f: V(G)\to [1,p]$ of $G$ is defined by $str_f(G) = \max\{f(u)+f(v)\; |\; uv\in
E(G)\},$ and the strength $str(G)$ of a graph $G$ itself is $str(G) =
\min\{str_f(G)\;|\; f \mbox{ is a numbering of } G\}.$ A numbering $f$ is
called a strength labeling of $G$ if $str_f(G)=str(G)$. In this paper, we
obtained a sufficient condition for a graph to have $str(G)=|V(G)|+\d(G)$.
Consequently, many questions raised in [Bounds for the strength of graphs, {\it
Aust. J. Combin.} {\bf72(3)}, (2018) 492--508] and [On the strength of some
trees, {\it AKCE Int. J. Graphs Comb.} (Online 2019)
doi.org/10.1016/j.akcej.2019.06.002] are solved. Moreover, we showed that every
graph $G$ either has $str(G)=|V(G)|+\d(G)$ or is a proper subgraph of a graph
$H$ that has $str(H) = |V(H)| + \d(H)$ with $\d(H)=\d(G)$. Further, new good
lower bounds of $str(G)$ are also obtained. Using these, we determined the
strength of 2-regular graphs and obtained new lower bounds of $str(Q_n)$ for
various $n$, where $Q_n$ is the $n$-regular hypercube.
