Salma Kanwal, Ayesha Riasat, Mariam Imtiaz, Zurdat Iftikhar, Sana Javed, Rehana Ashraf
November 15, 2018

### Abstract

A super edge-magic total (SEMT) labeling of a graph ℘( V , E ) is a one-one map ϒ from V (℘)∪ E (℘) onto {1, 2, … ,| V (℘)∪ E (℘) |} such that ∃ a constant “a” satisfying ϒ( υ ) + ϒ( υν ) + ϒ( ν ) = a , for each edge υν ∈ E (℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength , sm (℘), of a graph ℘ is the minimum of all magic constants a (ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as μs(℘)$\mu_{s}(\wp)$ is the least non-negative integer n so that ℘∪ nK 1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.