# 数学代写|图论作业代写Graph Theory代考|MATH361

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## 数学代写|图论作业代写Graph Theory代考|Connectivity and Flow

In each of the previous chapters, we have used connectivity in the context of other problems. For example, in Chapter 2 we needed to know if a graph was connected in order to determine if it is eulerian and in Chapter 3 we define trees as minimally connected graphs, since the removal of any edge would disconnect the graph. This chapter focuses on connectivity as its own topic, where we now consider how connected a graph is, and not just whether it is connected or not. Note that this chapter is more theoretical than the previous, though we tie in network flow and end with a section on applications of connectivity.

Consider graphs $G_1, G_2$, and $G_3$ below. It should be plain to see that they are all connected graphs. We could describe other features of these graphs (are they eulerian? hamiltonian? acyclic?) but one distinguishing factor between them should be the simple difference in edge count, with $G_1$ having fewer edges than the other two. Notice that $G_2$ and $G_3$ both contain 13 edges, but their underlying structure is quite different. Visually, is seems that the edges in $G_3$ are more clumped than in $G_2$. One way to describe this clumping is in how many edges or vertices would need to be removed before the graph is no longer connected, which is one way we measure connectivity.

## 数学代写|图论作业代写Graph Theory代考|Connectivity Measures

When we define a graph to be connected, we refer to the existence of a way to move between any two vertices in a graph, specifically as the existence of a path between any pair of vertices. We will see that measuring how connected a graph is has a similar description, but we first use the standard notion described above in terms of vertex (or edge) removal.

Definition $4.1$ A cut-vertex of a graph $G$ is a vertex $v$ whose removal disconnects the graph, that is, $G$ is connected but $G-v$ is not. A set $S$ of vertices within a graph $G$ is a cut-set if $G-S$ is disconnected.

Note that any connected graph that is not complete has a cut-set, whereas $K_n$ does not have a cut-set (see Exercise 4.13). Moreover, a graph can have many different cut-sets and of varying sizes. For example, two different cut-sets are shown below for graph $G_1$ above.

Although we can find many different cut-sets for graph $G_1$, we may want to choose one over another based on some sense of optimality. In particular, when we evaluate how connected a graph is, we are really asking what is the fewest number of vertices whose removal will disconnect the graph.

The distinction between $k$-connected and connectivity $k$ is subtle yet important. For example if we say a graph is 3 -connected, then we know there cannot be a cut-set of size 2 or less in the graph; however, we only know that its connectivity is at least $3(\kappa(G) \geq 3)$.

## 数学代写|图论作业代写图论代考|连通性度量

. 当我们定义一个要连接的图时，我们指的是图中任意两个顶点之间移动的方法的存在性，具体来说就是任意一对顶点之间的路径的存在性。我们将看到，测量一个图的连接程度有类似的描述，但我们首先使用上面描述的顶点(或边)去除的标准概念

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## MATLAB代写

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