Graph theory - basic concepts and data structures of graphs

Basic concept of graph

Undirected graph

Edges have no direction, that is, bidirectional

结点V = { v 1 , v 2 , . . . , v 7 } \mathcal{V} = \{ v_1,v_2,...,v_7\}V={ v1​,v2​,...,v7​}

边ε = { and 1 , 2 , and 1 , 3 , . . . , e 6 , 7 } \varepsilon = \{e_{1,2},e_{1,3},...,e_{6,7}\}e={ e1,2​,e1,3​,...,e6,7​}

图G = { V , ε } \mathcal{G} = \{ \mathcal{V},\varepsilon \}G={ V,e }

directed graph

Edges are directional, that is, unidirectional

Unweighted graph

The edge has no weight, which can also be understood as the weight is 1

entitled figure

Edges have weights

Graph Data Structure

Adjacency matrix and adjacency list

undirected unweighted graph

Adjacency list

Indicates who the node is connected to,

vertex	Neighbors
1	2,3,4
2	1,4
3	1,4,6
4	1,2,3
5	empty
6	3,7
7	6

Adjacency matrix

$$\left[\begin{array}{ccccccc}0& 1& 1& 1& 0& 0& 0\\ 1& 0& 0& 1& 0& 0& 0\\ 1& 0& 0& 1& 0& 1& 0\\ 1& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0\end{array}\right]$$

The adjacency matrix represents the nodes $i$ and nodejjThe weight between$j\; ,\; if\; it\; is\; an\; unweighted\; graph,\; the\; weight\; is\; 1.$The adjacency matrices of undirected graphs are all symmetric. Because edges are bidirectional.

directed unweighted graph

Adjacency list

Indicates who the node is connected to,

vertex	Neighbors
1	2,4
2	4,5
3	1,6
4	5
5	empty
6	7
7	6

Adjacency matrix

$$\left[\begin{array}{ccccccc}0& 1& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 0& 0\\ 1& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0\end{array}\right]$$

The adjacency matrix represents the nodes $i$ and nodejjThe weight between$j\; .$The list shows the starting point, and the row shows the reached point, such as$a_{1,2}=1$ means there is a unidirectional edge from node 1 to node 2.

entitled figure

Adjacency matrix

$$\left[\begin{array}{ccccccc}0& 2& 4& 1& 0& 0& 0\\ 2& 0& 0& 3& 0& 0& 0\\ 4& 0& 2& 0& 0& 5& 0\\ 1& 3& 2& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 5& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0\end{array}\right]$$

In fact, it is very similar to an undirected graph, which means the link between two nodes. The weighted graph has weights, but the unweighted graph does not.

degree matrix

The degree matrix is ​​an important concept in graph theory, which is used to describe the degree of each node in an undirected or directed graph. It is a diagonal matrix, the elements $d_i$represents node $degree\; of\; i$ , i.e. with node$i$ is the number of connected edges.

Undirected graph

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
$$\left[\begin{array}{ccccccc}3& 0& 0& 0& 0& 0& 0\\ 0& 2& 0& 0& 0& 0& 0\\ 0& 0& 3& 0& 0& 0& 0\\ 0& 0& 0& 3& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$
It can be seen that is a diagonal matrix.

directed graph

For directed graphs, it can be divided into out-degree matrix and in-degree matrix. The out-degree matrix represents the out-degree of each node, and the in-degree matrix represents the in-degree of each node. This is easy to calculate, not in the demonstration

Laplacian matrix

The Laplacian matrix is ​​used to describe the properties of graphs. There are generally several methods for calculating the Laplacian matrix

1. Symmetric normalized Laplacian: Let L be the Laplacian matrix of graph G, D be the degree matrix, then the symmetric normalized Laplacian matrix is $L_{sym}=D^{-\frac{1}{2}}L{D}^{-\frac{1}{2}}$。

2、Random walk Laplacian：设 $M=D^{-1}L$, then the random walk Laplacian matrix is$L_r=I-M$ ,$I$ is the identity matrix.

3. Unnormalized Laplacian: The Laplacian matrix without normalization can be expressed as $L=D-A$ , where A is the adjacency matrix and D is the degree matrix

Matrix properties[1]

• 0 is his eigenvalue, $1_N$is the eigenvector

• $x^T\ell x=\frac{1}{2}{\sum }_{i=1}^N{\sum }_{j=1}^N{a}_{ij}(x_j-x_i{)}^2$，ℓ \heThe positive semi-definiteness of$\ell$$All\; characteristic\; roots\; of\; \ell$ are real and nonnegative

• If G is connected, then ℓ \ellThe second smallest characteristic root of$\ell$$l_2\left(\ell \right)$ , called the algebraic connectivity of G, which is greater than 0

• The algebraic connectivity of G is equal to $mi{n}_{x\ne 0,1_N^{T}x=0}\frac{x^T\ell x}{x^{T}x}$, so if $1_N^T x = 0,x^T \ell x \ge \lambda_2(\ell)x^Tx $ .

references

[1] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004)

The pictures in this article are from the course of Mr. Wang Shusen. The course has not been carefully studied, and only these basic knowledge are needed.