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# 数据科学代写|复杂网络代写Complex Network代考|ACM40970 Representations of graphs

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## 数据科学代写|复杂网络代写Complex Network代考|Representations of graphs

By definition, a graph is represented by a set of pairs of labels; each pair contains two labels of the end vertices of the corresponding edge. For a directed edge, this pair is ordered. There are two other convenient ways to mathematically represent a graph: (i) an incidence matrix indicating the pairs: an edge – a vertex incident to it, and (ii) an adjacency matrix indicating the pairs of adjacent vertices.

The incidence matrix $B$ of a graph with $N$ vertices and $E$ edges has $N$ rows and $E$ columns. For an oriented graph, ${ }^{11}$ this $N \times E$ matrix has the elements: ${ }^{12}$

• $B_{i j}=-1$ if vertex $i$ is the source vertex of edge $j$;
• $B_{i j}=+1$ if vertex $i$ is the target vertex of edge $j$;
• $B_{i j}=2$ if edge $j$ is a self-loop at vertex $i$;
• the remaining entries are zero.
So, for a simple oriented graph, the sum of elements in each column of this matrix is zero, $\sum_{i} B_{i j}=0$ (an edge has one source and one target).

The adjacency matrix of an undirected graph of $N$ vertices, $A(G)$, can be constructed in the following way. Start with the $N \times N$ matrix having all its entries zeroes. For each edge between vertices $i$ and $j$ in the graph, add 1 to the entries $A_{i j}$ and $A_{j i}$ of the matrix, and for each self-loop at vertex $i$ add 2 to the entry $A_{i i}$ of the matrix. The resulting adjacency matrix of an undirected graph of $N$ vertices is a $N \times N$ symmetric matrix with the entries:

• $A_{i j}=n$ if there are $n$ edges interconnecting vertices $i$ and $j, i \neq j$
• $A_{i i}=2 m$ if vertex $i$ has $m$ self-loops;
• the remaining entries are zero.
In particular, in the adjacency matrix of a simple undirected graph, $A_{i j}=1$ if there is an edge between vertices $i$ and $j, A_{i j}=0$ if such edge is absent, and the diagonal elements are zero, $A_{i i}=0$.

The adjacency matrix of an oriented graph of $N$ vertices can be introduced in a similar way. Start with the $N \times N$ matrix having all its entries zeroes. For each edge going from vertex $i$ to vertex $j$ in the graph, add 1 to the entry $A_{i j}$ of the matrix, and for each self-loop at vertex $i$ add 1 to the entry $A_{i i}$ of the matrix. ${ }^{14}$ The resulting adjacency matrix of an oriented graph of $N$ vertices is a $N \times N$ matrix with the entries:

• $A_{i j}=n$ if there are $n$ edges going from vertex $i$ to vertex $j, i \neq j$;
• $A_{i i}=m$ if vertex $i$ has $m$ directed self-loops;
• the remaining entries are zero.

## 数据科学代写|复杂网络代写Complex Network代考|Representations of graphs

$B_{i j}=-1$ 如果顶点 $i$ 是边的源页点 $j$;

$B_{i j}=+1$ 如果顶点 $i$ 是边的目标顶点 $j$;

$B_{i j}=2$ 如果边豚 $j$ 是顶点处的自环 $i$;

$A_{i j}=n$ 如果有 $n$ 连接顶点的边 $i$ 和 $j, i \neq j$

$A_{i i}=2 m$ 如果顶点 $i$ 有 $m$ 自循环;

$A_{i j}=n$ 如果有 $n$ 从顶点开始的边 $i$ 到顶点 $j, i \neq j ;$

$A_{i i}=m$ 如果顶点 $i$ 有 $m$ 有向自环;

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