[Management Operations Research] Chapter 7 | Graph and Network Analysis (1, Graph theory background and basic concepts, terminology, matrix representation)

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introduction

According to the normal progress, I should learn dynamic programming, but I want to change my taste, and I heard that dynamic programming is also difficult, and I may not be able to take the exam.

Let me talk about some background knowledge and development of graph theory first.

Graph theory is a new branch of mathematics that has developed rapidly and is widely used in decades. It is closely related to other branches of mathematics such as matrix theory, probability theory, and numerical analysis. In fact, graph theory provides a mathematical model for any system involving a binary relationship; also because it uses a diagrammatic representation, the graph has an intuitive and aesthetic appearance.

The development of graph theory can be roughly divided into three stages.

The first stage, from the middle of the 18th century to the middle of the 19th century, is called the budding period. The origin is the "seven bridges game" problem, shown in the figure below.

The question is: is it possible to start from any of these four landmass, cross each bridge once, and only once, and return to the starting point again.

The Swiss mathematician Euler (Euler) published the first paper on graph theory on this issue, proving that there is no solution to the seven-bridge game problem, and further generalized this problem (edge-stroke problem), A judging rule for the existence of Euler circles in a graph is given.

Since the Chinese Postman Problem (Chinese Postman Problem) was proposed, the Euler problem has a strong practical value.

The problem of the Chinese postman is this: the postman must first pick up the mail he should distribute from the post office before starting along the postal route. In order to save time, every postman is willing to take all the routes he has to take with as few trips as possible. In terms of graph theory, it refers to how to traverse all the streets on the postal road with as few trips as possible and return to his starting point.

The first paper on this type of problem was proposed by Chinese mathematician Professor Guan Meigu of Shandong Normal University in 1962, hence the name "Chinese Postman Problem".

From the middle of the 19th century to the middle of the 20th century is the second stage of the development of graph theory. During this period, a large number of problems in graph theory emerged, among which the Hamilton problem and the four-color conjecture are the most famous.

In 1856, British mathematician Sir William Hamilton invented the game "Around the World". This game uses a regular dodecahedron, whose 20 vertices are marked with the names of 20 cities, and requires the player to find a route starting from a certain city, passing through each city exactly once, and finally returning to the starting point.

Project the dodecahedron onto the plane to get the following figure.

In fact, Hamilton's problem of traveling around the world is a point-and-stroke problem in graph theory. It is to find a circle $H$ : 1. Every vertex in the graph is in circle$Appear\; in\; H$ ; 2. in$The\; vertices\; in\; H$ do not appear repeatedly (the start and end points are not repeated). This circle is called a Hamilton circle.

The four-color conjecture problem, that is, whether it is possible to color the map with only 4 colors, so that neighboring countries have different colors. Using a graph to describe it is: use a point to represent a country, and if two countries have a common boundary, use an edge to connect the two points. Therefore, the four-color conjecture problem can be transformed into whether four colors can be used for the point of the plane. Coloring so that adjacent points have different colors.

After the middle of the 20th century, it was the third stage of the development of graph theory. During this period, graph theory experienced explosive development and grew into an independent discipline. The most important of these: the emergence of a powerful tool for research and problem solving: the computer.

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1. Basic knowledge of graphs and networks

1.1 Basic concepts of graphs and networks

1.1.1 Definition of graph

In nature and human society, a large number of things and the relationship between things can often be described by graphics. The research object is often regarded as a point, and a connection line (with or without arrows) is used to represent a certain relationship between objects. For the sake of distinction, we call the line without an arrow an edge , and the line with an arrow an arc .

Definition 1.1 — A graph is composed of a non-empty set $V$ , andVVA set EEconsisting of unordered (or ordered) pairs of elements in$V$$E$ (or$A$ ) composed of. VVThe set of unordered point pairs of elements in$V$$E$ , from the set of points$V$ and edge setEEThe graph composed of$E$ is called an undirected graph(referred to as a graph), which is recorded as$G=(V,E)$ . A connection point$v_i,v_j$The edge $e_{ij}$ ，记为 $e_{ij}=[v_i,v_j]$ 或 $e_{ij}=[v_j,v_i]$。VVThe set of ordered point pairs of elements in$V$$A$ , by point set$V$ and arc setAAThe graph composed of$A$ is a directed graph, recorded as$D=(V,A)$ . One direction is from$v_i$point to $v_j$The arc of $a_{ij}=(v_i,v_j)$ 。

Ruotu $In\; G$ , if the two endpoints of a side are the same, the side is called a ring, and if there is more than one side between two points, it is called a multiple side. A graph that is acyclic and has no multiple edges is calleda simple graph, and a graph that is acyclic but allows multiple edges is calleda multigraph.

Figure $G$ orDDThe number of points in$D$$n=|V|$ , the number of sides (arcs) is recorded as$m=|E|(m=|A|)$ , in the case of not causing confusion, they are abbreviated as$n,m$ , where$n$ is the order of the graph, if$n$ is finite and is called finite order.

1.1.2 Related terms in the figure

1. endpoint. This $e_{ij}=[v_i,v_j]$ , with side$e_{ij}$The connected vertices are called edges $e_{ij}$endpoint.
2. Edges are associated with points. When $e_{ij}=[v_i,v_j]$时，$e_{ij}$与$v_i,v_j$It is called side-vertex association.
3. neighbor.
4. adjacent side.
5. ring. Edges associated with only one vertex are called cycles.
6. parallel sides. Edges that have the same two endpoints are called parallel edges.
7. Area. A collection of points adjacent to a point.
8. Second-rate. Take point $v_i$The number of sides that are endpoints is called the point $v_i$in GGThe order in$G$$d(v_i)$ .
   If there is a cycle, record according to two sides, namely$$d(v_i)=d_l(v_i)+2l\left(v_i\right)$$ in which:$d_l\left(v_i\right)$ is the same as$v_i$The number of associated acyclic edges, $l(v_i)$ is the same as$v_i$The number of associated rings.
9. order sequence. If V = { v 1 , v 2 , … , vp } V=\{v_1,v_2,\dots,v_p\}V={ v1​,v2​,…,vp​} , then there is a degree relative to each point, you can get a degree sequence$(d(v_1),d(v_2),\dots )$ 。

Theorem 1.1 - For graph $G=(V,E)$ , where$|V|=n,|E|=m$ , then:$$\sum _{v\in V}d(v)=2m$$ Theorem 1.2—— The total number of vertices with odd times is even.

10. hanging point. times 1 point.
11. Overhang. The edge associated with the dangling point.
12. isolated point. The point where the time is 0.
13. chain.
14. elementary chain. Chain QQThe vertices in$Q\; are\; all\; different.$
15. simple chain. The edges in the chain are all different.
16. the length of the chain. is the number of sides included.
17. lock up.
18. road.
19. path. Each vertex of the road in a directed graph is different and called a path.
20. circuit. The first point of the road is the same as the last point.

1.1.3 Some special graph classes

1. Trivial graph. Number of nodes $n=1$ , number of sides$m=0$ graph.
2. Zero map. Number of sides $m=0$ 。
3. connected graph. Every pair of nodes in the graph is connected by a link (road), and the graph is said to be connected.
4. Tree. A connected graph without cycles.
5. complete graph. There is exactly one edge between any two vertices.
6. binary picture.
7. complete bipartite graph.
8. regular graph. Each point has the same number of times.
9. directed network. Weighted directed graph.

1.1.4 Graph operations

(1) Subgraph and support
Subgraph and support subgraph are all graphs GGIt is obtained by deleting the vertices or edges of$G.$Both subgraph points and edges are subsets of the original graph, supporting subgraph points are the same as the original graph, and edges are a subset of the original graph.

(2) Contraction operation of graph

(3) Cut sets
are often recorded as $\Phi \left(X\right)$ , as shown in the figure below, ifX = { V 1 } X=\{V_1\}X={ V1​} , Separation collection$\Phi \left(X\right)=\{[v_1,v_2],[v_1,v_3],[v_1,v_4\}$ . That is, to cut with a line, it is required to cut$X$ is completely cut out, and the edges touched by this line are recorded as cut sets.

(4) Isomorphism of graphs
Let $G_1,G_2$For two graphs of the same order, if the vertex set $V_1,V_2$And edge set $E_1,E_2$There is a one-to-one correspondence between them under the condition of maintaining the correlation property, then the graph $G_1,G_2$isomorphic. As shown below.

1.2 Matrix representation of graph

In order to facilitate the use of computers to identify the relevant information of the graph, we use matrix representation to represent a graph, mainly including adjacency matrix, incidence matrix, reachability matrix, weight matrix, etc.

1.2.1 Adjacency matrix

The adjacency matrix is ​​used to describe whether two vertices are connected by an edge (arc). If there is an edge (arc) between two points, the corresponding matrix element is 1, otherwise the corresponding matrix element is 0.

Obviously, the adjacency matrix of an undirected graph is a symmetric matrix about the diagonal.

1.2.2 Accessibility matrix

In a directed graph, the reachability matrix is ​​used to describe whether there is a path between two points. If there is a path, the corresponding matrix element is 1, otherwise it is 0.

1.2.3 Incidence Matrix

The incidence matrix of a directed graph is also known as the vertex-edge incidence matrix. Each line represents a point $v_i$, each column represents an arc $a_j$，Young $v_i$is the arc $a_j$The starting point of the corresponding matrix element $m_{ij}=1$ ; if it is the end point of the arc, it is recorded as -1, and if it is irrelevant, it is recorded as 0.

1.2.4 Weight Matrix

It can be understood as the extension of the adjacency matrix. When there is an edge or arc between two points, the corresponding matrix element is the weight of the edge or arc; when there is no edge between the two points, the corresponding element is $\infty$ ; The diagonal elements of the weight matrix are all 0.

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write at the end

There are really many concepts, but it is okay to understand them in combination with graphs. Later, we will talk about some classic problems in graph theory.