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# 数学代写|离散数学代写Discrete Mathematics代考|MA2201/CS2022 Incidence and Adjacency Matrices of a Graph

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## 数学代写|离散数学代写Discrete Mathematics代考|Incidence and Adjacency Matrices of a Graph

In this section we are assuming that our graphs are finite, directed, without loops, and without parallel edges. More explicitly, these directed graphs $G=(V, E)$ have the property that $E \subseteq V \times V$, and they have no edges of the form $(v, v)$.

Definition 10.9. Let $G=(V, E)$ be a graph and write $V=\left{\mathbf{v}1, \ldots, \mathbf{v}_m\right}$ and $E=$ $\left{\mathbf{e}_1, \ldots, \mathbf{e}_n\right}$. The incidence matrix $D(G)$ of $G$ is the $m \times n$-matrix whose entries $d{i j}$ are
$$d_{i j}= \begin{cases}+1 & \text { if } \mathbf{v}_i=s\left(\mathbf{e}_j\right) \ -1 & \text { if } \mathbf{v}_i=t\left(\mathbf{e}_j\right) \ 0 & \text { otherwise }\end{cases}$$
Remark: The incidence matrix actually makes sense for a graph $G$ with parallel edges but without loops.

For simplicity of notation and when no confusion is possible, we write $D$ instead of $D(G)$.

Because we assumed that $G$ has no loops, observe that every column of $D$ contains exactly two nonzero entries, $+1$ and $-1$. Also, the $i$ th row of $D$ is the vector $\omega\left(\mathbf{v}_i\right)$ representing the cocycle $\Omega\left(\mathbf{v}_i\right)$. For example, the incidence matrix of the graph $G_8$ shown again in Figure $10.17$ is shown below.

The incidence matrix $D$ of a graph $G$ represents a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ called the incidence map (or boundary map) and denoted by $D$ (or $\partial$ ). For every $e \in E$, we have
$$D\left(\mathbf{e}{\mathbf{j}}\right)=s\left(\mathbf{e}{\mathbf{j}}\right)-t\left(\mathbf{e}_{\mathbf{j}}\right)$$

## 数学代写|离散数学代写Discrete Mathematics代考|Network Flow Problems; The Max-Flow Min-Cut Theorem

The network flow problem is a perfect example of a problem that is important practically but also theoretically because in both cases it has unexpected applications. In this section, we solve the network flow problem using some of the notions from Sections 10.1-10.3. First, let us describe the kinds of graphs that we are dealing with, usually called networks (or transportation networks or flow networks).

Definition 10.12. A network (or flow network) is a quadruple $N=\left(G, c, v_s, s_t\right)$, where $G$ is a finite digraph $G=(V, E, s, t)$ without loops, $c: E \rightarrow \mathbb{R}_{+}$is a function called a capacity function assigning a capacity $c(e)>0$ (or cost or weight) to every edge $e \in E$, and $v_s, v_t \in V$ are two (distinct) distinguished nodes. ${ }^3$ Moreover, we assume that there are no edges coming into $v_s\left(d_G^{-}\left(v_s\right)=0\right)$, which is called the source and that there are no outgoing edges from $v_t\left(d_G^{+}\left(v_t\right)=0\right)$, which is called the terminal (or $\sin k$ ).

10.9.

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