On Two Conjectures in Mod Sum Graphs

In this paper we consider the two conjectures in mod sum graphs, put forward by J. Bolland, R. Laskar, C. Turner and G. Domke in [2]. We prove one of the conjectures and disprove the other. Namely, we prove that every connected graph can appear as a subgraph of a connected mod sum graph; and we show that no wheel with the exception of W 4 , can be a mod sum graph. We also give the unique (up to multiplication by positive integers) mod sum labelling of W 4 . Keywords: mod sum labelling, mod sum graphs, sum graphs, sum graph labelling, wheels. 1 Introduction A sum graph is a graph G = (V; E) and a labelling of the vertices of G with distinct elements of the finite subset S ae Z + such that v i v j 2 E if and only if the sum of the labels assigned to v i and v j is the label of a vertex of G. It is obvious that a sum graph cannot be connected. There must always be at least one isolated vertex, namely the vertex with the highest label. The sum number oe(H) of a connected graph H is th..