# 数学代写|图论作业代写Graph Theory代考|MAT6495

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

As we have already seen, we measure connectivity of a graph in terms of the number of distinct paths between two vertices or by the number of vertices (or edges) needed to disconnect the graph. This section will investigate an-other method for evaluating the underlying structure of a graph by identifying central vertices.

In Definition 2.29, we define a vertex to be central if its eccentricity equals the radius of the graph. Recall the eccentricity of a vertex $x$ is the maximum distance from $x$ to any other vertex in $G$; that is $\epsilon(x)=\max {y \in V(G)} d(x, y)$ and the radius of a graph is the minimum eccentricity among all vertices; that is $\operatorname{rad}(G)=\min {x \in V(G)} \epsilon(x)$. These are useful measures for identifying vertices within short distance of most other vertices, but does that really explain how connected or important a single vertex is to the rest of the graph?

Instead of only relying on path distance, we may want to characterize vertices based on other metrics indicating their relative importance within a graph. These measures are often called network centralities, where network here simply means a connected graph.

Consider, for example, your network of friends. If each person were a vertex and an edge represented a friendship, then some of your friends would naturally be friends with each other, and you might find clumps in the vertices, maybe your math major friends, intermural soccer friends, or friends from your dorm; some of these clumps may overlap, so you are not the only connection from these groups of people.

We will use graph $G_{10}$ above as an example for the various centrality measures. However, in most applications we would be working with graphs containing many more vertices (think in the range of a few hundred to many millions) and our analysis would be done using computer software and the adjacency matrix; one such example is shown at the conclusion of this section.
Using a friend network as motivation, how might we identify a central vertex? Perhaps the person with the most friends? This is called the degree centrality.

## 数学代写|图论作业代写Graph Theory代考|Matching and Factors

In Chapter 4, we focused on the underlying structure of a graph in terms of its connectivity. In this chapter we again focus on the structure of the graph but from the viewpoint of grouping vertices based on a variety of criteria, mainly in terms of making viable pairings. In the next chapter, we will again group vertices (or edges) together in order to avoid conflict. To begin this chapter, we investigate the optimization of pairings through the use of edge-matchings within a graph, more commonly known as a matching.

Definition 5.1 Given a graph $G=(V, E)$, a matching $M$ is a subset of the edges of $G$ so that no two edges share an endpoint. The size of a matching, denoted $|M|$, is the number of edges in the matching.

Recall that when two edges do not share an endpoint, we call them independent edges, so a matching is just a set of independent edges within a graph.

The most common application of matchings is the pairing of people, usually described in terms of marriages. Other applications of a graph matching are task assignment, distinct representatives, and roommate selection.

Example 5.1 The Vermont Maple Factory just received a rush order for 6-dozen boxes of maple cookies, 3-dozen bags of maple candy, and 10dozen bottles of maple syrup. Some employees have volunteered to stay late tonight to help finish the orders. In the chart below, each employee is shown along with the jobs for which he or she is qualified. Draw a graph to model this situation and find a matching.

## 数学代写|图论作业代写图论代考|匹配与因子

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