Question 1 Consider the following directed graph G(V, E):
V2
e1
e4
V3
e5
V1
e6
e3
e2
e7
V4
V5
(1.1) Find the adjacency matrix A, incidence matrix B, and C as the adjacency matrix of the undirected version of the graph.
(1.2) Show that $B^T \mathbf{1} = \mathbf{0}$, where $\mathbf{1}$ and $\mathbf{0}$ represent the all-ones and zero vectors, respectively.
(1.3) Based on 1.2, find an eigenvalue and eigenvector of $B^T B$ and $BB^T$.
(1.4) Let $L = BB^T$. Show that $L = D - C$, where $D = \text{diag}(\text{deg}(v_1), ..., \text{deg}(v_5))$.
(1.5) Find $\text{det}(L)$.
(1.6) Show that all eigenvalues of $L$ are nonnegative.
[1.1:3, 1.2:1, 1.3:8, 1.4:2, 1.5:3, 1.6:3=20 marks]
