[Machine Learning | Mathematical Fundamentals] Graph Theory of Mathematics for Machine Learning Series (12): Directed Euler Graph

foreword

Hello! Buddy!
Thank you very much for reading Haihong's article. If there are mistakes in the text, you are welcome to point out ~
 
self-introduction ଘ(੭ˊᵕˋ)੭
Nickname: Haihong
Label: Programmer | C++ Player | Student
Profile: I got acquainted with programming in C language, and then transferred to Entered the computer science major, and was fortunate to have won some national awards and provincial awards... Has been guaranteed. Currently learning C++/Linux/Python
learning experience: solid foundation + more notes + more typing + more thinking + learning English!
 
Machine learning novice stage
The article is only used as your own study notes for knowledge system establishment and review
. Know it and know why!

series of articles

[Machine Learning | Mathematical Fundamentals] Graph Theory of Mathematics for Machine Learning Series (1): Basic Concepts of Graphs

[Machine Learning | Mathematical Fundamentals] Graph Theory of the Mathematics for Machine Learning Series (2): Matrix Representation of Graphs

[Machine Learning | Mathematical Fundamentals] Graph Theory of the Mathematics for Machine Learning Series (3): Paths and Connections

[Machine Learning | Mathematical Fundamentals] Graph Theory of the Mathematics for Machine Learning Series (4): Connectivity of Directed Graphs

[Machine Learning | Mathematical Fundamentals] Graph Theory of Mathematics for Machine Learning Series (5): Trees and Their Properties

[Machine Learning | Mathematical Fundamentals] Graph Theory of the Mathematics for Machine Learning Series (6): Spanning Tree Algorithm

[Machine Learning | Mathematical Fundamentals] Graph Theory of the Mathematics for Machine Learning Series (7): Connectivity

[Machine Learning | Mathematical Fundamentals] Graph Theory of Mathematics for Machine Learning Series (8): Cut Edges, Cut Sets, Cut Points

[Machine Learning | Mathematical Fundamentals] Graph Theory of Mathematics for Machine Learning Series (9): The Concept of Matching

[Machine Learning | Mathematical Fundamentals] Graph Theory of Mathematics for Machine Learning Series (10): Matching Fundamental Theorem

[Machine Learning | Mathematical Fundamentals] Graph Theory of Mathematics for Machine Learning Series (11): Euler Graph

6.3 Directed Euler Graphs

Scheduled for 6.2

Let $G=(V,E)$ is a weakly connected directed graph

directed tour

through GGA directed closed path at least once for each arc of$G\; is\; called\; a\; directed\; itinerary$

Directed Euler Tour

through GGA directed tour that occurs exactly onceper arc of$G$ is called a directed Euler tour

Directed Euler graph

A directed graph with a directed Euler tour is called a directed Euler graph

Directed Euler path

through $A\; directed\; path\; where\; each\; arc\; of\; G$ is exactly once is called a directed Euler path

Theorem 6.4

$G$ is a weakly connected directed graph, then the following propositions are equivalent

1. $G$ is a directed Euler graph
2. $\forall v\in V\left(G\right),d^{-}\left(v\right)=d^{+}\left(v\right)$
3. $G={\bigcup }_{i=1}^n{C}_i$，$C_i$is a directed cycle, and $E\left(C_i\right)\cap E\left(C_j\right)=\varphi ,i\le i<j\le n$，$n$ is some natural number

Corollary 6.4

Directed circle $G$ Yes$u_1$as the starting point, take $u_2$The necessary and sufficient conditions for a directed Euler path to the end point are: $G$ is a weakly connected directed graph that satisfies

Epilogue

illustrate:

• Refer to the textbook "Graph Theory"
• Cooperate with the concept explanation in the book and combine some of my own understanding and thinking

The article is only used as a study note, recording a process from 0 to 1

I hope it can help you a little bit, if there are any mistakes, please correct me