(Solved): Let \( \mathbf{G}=\mathbf{( V}, \mathbf{E}) \) be a simple graph with \( \mathbf{n}=|\mathbf{V}| \ ...

Let \( \mathbf{G}=\mathbf{( V}, \mathbf{E}) \) be a simple graph with \( \mathbf{n}=|\mathbf{V}| \) vertices, and let \( \mathbf{A} \) be its adjacency matrix of dimension \( \mathbf{n} \times \mathbf{n} \). We want to count the \( \mathrm{I} \)-cycles: such a cycle, denoted by \( \mathrm{C}=\mathrm{u}_{0} \mathrm{u}_{1} \cdot \mathrm{u}_{1} \) with \( u_{1}=u_{0} \) contains \( / \) distinct vertices \( u_{0}, \ldots, u_{1-1} \), and \( / \) edges \( E(C)=\left\{u_{i} u_{i+1} \mid 0 \leq i \leq I-1\right\} \) \( \subseteq \mathrm{E} \). Two cycles are distinct if the sets of edges are different: \( C=C^{\prime} \) if and only if \( E(C)= \) \( E\left(C^{\prime}\right) \). We define the matrices \( D, T, Q \), the powers of \( A \) by matrix multiplication: \[ D=A \cdot A=A^{2} \quad T=A \cdot D=A^{3} \quad Q=A \cdot T=A^{4} \] Consider the values on the diagonals. 1. Prove that the number of 3-cycles \( \mathrm{N} 3 \) in the entire graph is \[ N_{3}=\frac{1}{6} \sum_{u \in V} T_{u, u} \] 2. Develop a formula to calculate the number of 4-cycles \( \mathbf{N} 4 \) from the diagonal values in the matrixes \( Q, T \) and \( D \). Justify the answer.