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# 数学代写|图论代写Graph Theory代考|EXTENSIONS

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## 数学代写|图论代写Graph Theory代考|Directed graphs

In the graphs we have considered so far, two vertices could be connected by one or more edges. An edge was represented by an unordered pair of vertices, such as $\langle u, v\rangle$ in the case of simple graphs. However, having no ordering is not always convenient. Consider the following examples:

• Suppose we want to model a street plan as a network. This is naturally done by representing a junction as a vertex and a street as an edge connecting two junctions. However, we need a notion of edge direction if we want to represent one-way streets.
• In social relations it is often convenient to represent the fact that Alice knows Bob, but that the opposite is not the case. In a social network this is done by representing people by vertices, and the “who knows whom” relation by means of directed edge.
• In computer networks, and notably wireless networks, links between two different nodes are often not symmetric in the sense that messages can generally be successfully sent from station $A$ to $B$, but not the other way around. Modeling such a computer network is more conveniently done using directed edges.

What we are thus seeking is a way to extend graphs that we will be able to model these and similar situations.

## 数学代写|图论代写Graph Theory代考|Weighted graphs

Let us now direct our attention to another important extension of the foundations discussed in Chapter 2 , namely assigning weights to edges (or arcs). A weight is a real-valued number associated with an edge. This extension is a natural one when modeling real-world networks as graphs. For example, when modeling a railway network as a graph, railway stations are naturally represented by vertices, whereas two adjacent stations are connected by means of an edge. We then assign a weight to an edge representing the distance between those two stations.

Definition 3.6: A weighted graph $G$ is a graph for which each edge e has an associated real-valued number w(e) called its weight. For any subgraph $H \subseteq G$, the weight of $H$ is simply the sum of weights of its edges: $w(H)=\sum_{e \in E(H)} w(e)$.

## Matlab代写

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