Super Edge-magic Total Strength of Some Unicyclic Graphs

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 V (G) E (G) \1, 2, , p+q\\) such that for all edges \ (uv E (G) \), \ (f (u) + f (v) + f (uv) = c (f) \), where \ (c (f) \) is called a magic constant, and \ (f (V (G) ) = \1, , 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 evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic \ ( (p, q) \) -graphs is equal to \ (2q + n+32\).