3. The matrix representation of the graph

Suppose G is any undirected graph that is formed by two finite sets V and E, which are called the vertices and edges, respectively. In other words, � � <sup>V</sup> <sup>¼</sup> <sup>f</sup>v1; <sup>v</sup>2; …; vl<sup>g</sup> and <sup>E</sup> <sup>¼</sup> <sup>f</sup>e1;e2; …;emg: The matrix representation A Gð Þ¼ eij <sup>l</sup>�<sup>m</sup> on graph G has been defined by

$$A(G) = \begin{bmatrix} \upsilon\_1 \\ \upsilon\_2 \\ \vdots \\ \upsilon\_l \\ \upsilon\_{l\_l} \end{bmatrix} \begin{bmatrix} \boldsymbol{e}\_{1\_1} & \boldsymbol{e}\_{2\_1} & \boldsymbol{e}\_{3\_1} & \dots & \boldsymbol{e}\_{m\_1} \\ \boldsymbol{e}\_{1\_2} & \boldsymbol{e}\_{2\_2} & \boldsymbol{e}\_{3\_2} & \dots & \boldsymbol{e}\_{m\_2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \boldsymbol{e}\_{1\_l} & \boldsymbol{e}\_{2\_l} & \boldsymbol{e}\_{2\_l} & \dots & \boldsymbol{e}\_{m\_l} \end{bmatrix} \tag{1}$$

� � with l rows corresponding to the l vertices vi and the m columns corresponding to the m edges ei: Whereas the incidence matrix of a connected digraph can be defined by A ¼ eij , where eij ∈ f0;∓ 1g: In other words, if j thedge is incident <sup>l</sup>�<sup>m</sup> out of i th vertex, then eij <sup>¼</sup> <sup>1</sup>, while eij ¼ �1, if <sup>j</sup> th edge is incident into i th vertex and if j th edge is neither incident out nor incident into i th vertex, then eij <sup>¼</sup> <sup>0</sup> [16, 17].
