2. The vector representation of the graph

Suppose G is a graph as shown in Figure 1.

A graph G has four vertices and five edges e1, e2, e3, e4, and e5: A subgraph H (and any other subgraphs) of G is represented by a 5-tuple.

This means that E ¼ ðe1;e2;e3;e4;e5Þ such that

$$e\_i = \mathbf{1}, \text{ if } e\_i \text{ is in } H,$$

$$e\_i = 0,\text{ if } e\_i \text{ is not in } H.$$

The subgraphs H<sup>1</sup> and H<sup>2</sup> in Figure 1 can be represented by (1,0,1,0,1) and (0,1,1,1,0), respectively. Here, there are 25 = 32 possible cases for 5-tuples which

Figure 1. The subgraphs H<sup>1</sup> and H<sup>2</sup> of the graph G [16]. correspond to 32 subgraphs. Among them are the (0,0,0,0,0) and (1,1,1,1,1) which represent a null graph and a graph G itself, respectively [16].
