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# 数学代写|图论代考GRAPH THEORY代写|MS-E1050 Complete Graphs

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## 数学代写|图论代写GRAPH THEORY代考|Complete Graphs

The weighted graph in Example 1.7 and $G_3$ from page 2 have the same underlying structure. If we remove the weights of edges and the vertex names, we would not be able to distinguish between the two graphs since in both graphs every pair of distinct vertices is joined by an edge. These graphs are called complete graphs.

Definition 1.9 A simple graph $G$ is complete if every pair of distinct vertices is adjacent. The complete graph on $n$ vertices is denoted $K_n$.
The first six complete graphs are shown on the next page.

Complete graphs are special for a number of reasons. In particular, if you think of an edge as describing a relationship between two objects, then a complete graph represents a scenario where every pair of vertices satisfies this relationship. Other useful properties of complete graphs are given below.
Properties of $K_n$
(1) Each vertex in $K_n$ has degree $n-1$.
(2) $K_n$ has $\frac{n(n-1)}{2}$ edges.
(3) $K_n$ contains the most edges out of all simple graphs on $n$ vertices.

## 数学代写|图论代写GRAPH THEORY代考|Graph Complements

Consider a graph representing friendships. Given a collection of people, we could form a graph where an edge exists between two vertices if those people are friends. But what, if instead, we want to know who are not friends with each other? Perhaps a teacher wants to avoid friends talking during class and so will not seat them at the same table. This new graph would include all the edges missing from the original graph created, and is called the graph complement.

Definition 1.11 Given a simple graph $G=(V, E)$, define the complement of $G$ as the graph $\bar{G}=(V, \bar{E})$, where an edge $x y \in \bar{E}$ if and only if $x y \notin E$.

Note that graph complements are only defined for simple graphs (graphs without loops and multi-edges).

## 数学代写|图论代与GRAPH THEORY代考|Complete Graphs

(1) 中的每个顶点 $K_n$ 有学位 $n-1$.
(2) $K_n$ 有 $\frac{n(n-1)}{2}$ 边缘。
(3) $K_n$ 包含所有简单图形中最多的边 $n$ 顶点。

## MATLAB代写

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