Definitions

To begin, we first must define a simple undirected graph. A simple undirected graph $G=(V,E)$ consists of a set $V$ which is a nonempty, finite set of elements called vertices, and a set $E$ which is a set of unordered pairs $\{u,v\}$ called edges, where $u,v$ are distinct elements of $V$. $V$ is called the vertex set of $G$, and $E$ is called the edge set of $G$. A simple graph can have at most one edge between any two vertices. From now on we will use the word graph to mean a simple undirected graph. We now define some classes of graphs. The complete graph, $K_n$, has n vertices and every pair of vertices share an edge. The complete m-partite graph, $K_{n_1,n_2,...,n_m}$, has partite sets $V_1,V_2,...,V_m$ with $\vert V_i \vert = n_i$ having the property that if $u \in V_i$, $v \in V_j$, $i \ne j$ then $\{u,v\} \in E$ and $\{u,v\} \not \in E$ when $i=j$ . One class of complete m-partite graphs is $H_{n,m}$ which is a m-partite graph which each partite set containing $n$ vertices. A cycle containing n vertices is denoted $C_n$. Also a wheel $W_n$ contains a cycle of length n and an additional vertex which is adjacent to every vertex on the cycle. A tree $T_n$ is a acyclic connected graph containing $n$ vertices, while a forest $F_n$ is a acyclic graph containing $n$ vertices. The concept of sum graph was introduced by Harary at the Nineteenth Southeastern Conference on Combinatorics at Baton Rouge in 1988. A graph, $G=(V,E)$, is a sum graph if there exist a set $\cal F \subset \hbox{Z\kern-.35em\hbox{Z}}^+$ and a bijection $f: {\cal {F}} \rightarrow V$ such that if $x,y$ are distinct elements of $\cal F$, then $x+y \in \cal F$ if and only if $\{f(x),f(y) \} \in E$. In this paper, we will refer to vertices by their corresponding label in $\cal F$. The first obvious class of graphs which are not sum graphs are connected graphs. Then not all graphs are sum graphs and so it is a natural extension to define sum number of a graph. The sum number of graph $G$, denoted $\sigma(G)$, is the smallest number of isolated vertices that can be added to $G$ to yield a sum graph.