A sum labelling of the crown graph and some families of graphs

In 2008, H. Fernau et.al provided an optimal sum labelling scheme of the generalized friendship graph and showed that its sum number is 2. The generalized friendship graph fq p is a symmetric collection of cycles meeting at a common vertex. This graph may also be viewed as a graph obtained by considering several copies of a cycle and identifying a vertex from each cycle and merging them into a single vertex. In this paper, we consider a cycle and several paths and form a graph by concatenating a pendant vertex from a path to a vertex in the cycle. We also determine the exact value or a bound for the sum number of the resulting graph. Specifically we show that the sum number of the ladder graph P2 Pn is at most n, the tadpole graph Tn m and the graph SmCn is at most 2 and that the crown graph Ckn has a 1-optimal sum labelling