  Let $G$ be a finite simple undirected $(p,q)$-graph, with vertex set $V(G)$
and edge set $E(G)$ such that $p=|V(G)|$ and $q=|E(G)|$. A super edge-magic
total labeling $f$ of $G$ is a bijection $f\colon V(G)\cup E(G)\longrightarrow
\{1,2,\dots , p+q\}$ such that for all edges $u v\in E(G)$, $f(u)+f(v)+f(u
v)=c(f)$, where $c(f)$ is called a magic constant, and $f(V(G))=\{1,\dots ,
p\}$. The minimum of all $c(f)$, where the minimum is taken over all the super
edge-magic total labelings $f$ of $G$, is defined to be the super edge-magic
total strength of the graph $G$. In this article, we work on certain classes of
unicyclic graphs and provide shreds of evidence to conjecture that the super
edge-magic total strength of a certain family of unicyclic $(p,q)$-graphs is
equal to $2q+\frac{n+3}{2}$.
