A non-trivial undirected tree may be a hamilton graph
A non-trivial undirected tree could not be a hamilton graph. An undirected tree is a connected graph without a loop. Its property is the number of edges, m = n - 1 (n is the number of vertexes). The hamilton graph was a circuit that passed through all the points without overlapping. Undirected trees had no circuits, so a non-trivial undirected tree did not satisfy the definition of a hamilton graph. It could not be a hamilton graph. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
A non-trivial tree is an even graph
According to the definition, trees and forests were bigraphs. Non-trivial trees belonged to the category of trees, so non-trivial trees were bigraphs. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
A non-trivial tree must be a bi-graph
Yes, non-trivial trees are bigraphs. By definition, a bipartite graph meant that the set of vertexes could be divided into two disjoint sets, so that the two ends of each edge in the graph were in these two sets. For a non-trivial tree, it can be proved to be a bigram by marking its vertexes. With any one of the tree's vertexes as the root, the vertexes with even distances from the root were classified into a set, and the vertexes with odd distances from the root were classified into another set. Due to the structural characteristics of the tree, the two ends of any edge must be in these two sets respectively, satisfying the definition of a bigram. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
The relationship between undirected complete graph and Eulerian graph
An undirected complete graph was an undirected graph with edges connecting any two vertexes. An Eulerian graph was a graph that could be drawn in one stroke. In other words, there was a circuit that passed through all the edges and each edge only once. If the number of vertexes in an undirected complete graph was odd, then the undirected complete graph was an Eulerian graph. This was because an undirected complete graph with an odd number of vertexes could find such a single-stroke circuit. However, if the number of vertexes was even, it would not be an Eulerian graph. To put it simply, the odd or even number of vertexes of an undirected complete graph determined whether it was an Eulerian graph. It was an Eulerian graph with odd vertexes, but not with even vertexes. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
Judgement of Undirected Tree
The following was the method to determine an undirected tree: 1. ** Based on Connectedness and Circuit **: - If an undirected graph is connected and does not contain any simple circuits, it can be determined to be an undirected tree. 2. ** Based on the relationship between the number of edges and the number of vertexes (for the undirected connected graph of order n(n'geq2))**: - When the number of edges is (m = n - 1), there are at least two vertexes with degree (1) in the graph (G). At this time, it can be determined that the graph is an undirected tree. 3. ** Based on the uniqueness of the path between the vertexes **: - If the graph T is connected and there is only one simple path between each pair of different vertexes, then T is an undirected tree. 4. ** Based on the situation after the loop and edges are added **: - If there is no simple circuit in the graph, but there is a simple circuit in the graph obtained by adding a new edge between any two non-adjacent vertexes (i.e.,"maximum acyclic-free"), then "T" is an undirected tree. 5. ** Based on the necessity of the connection and edges **: - If the graph T is connected, but no longer connected after deleting any side (that is, every side in T is a bridge), then T is an undirected tree. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
The leaves of the undirected tree
In an undirected tree, the leaves referred to the apex with a degree of 1. Undirected trees have some characteristics, such as being connected and not containing any simple circuits. The number of edges m and the number of vertexes n (n>=2) satisfy m = n - 1. According to the relevant theorem, a non-trivial undirected tree has at least two leaves (a point with a degree of 1). In an undirected tree, except for the leaves, the apex with a degree greater than 1 was called a branch point. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
Every side of a non-ordinary undirected tree is a cut edge
A non-trivial undirected tree has the property that deleting any edge will make the graph no longer connected. According to the definition of cutting edge, if the edge is an edge of the graph, then when the number of connected branches is the number of connected branches, the edge is called the cut edge of the graph. For a non-trivial undirected tree, because removing any edge would no longer be connected, it meant that the number of connected branches increased after removing an edge. Therefore, every edge in the non-trivial undirected tree satisfied the definition of cutting edges, that is, every edge in the non-trivial undirected tree was a cut edge. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
It's neither an Eulerian graph nor a hamilton graph
The key to an Eulerian graph was to traverse all the edges without repeating them (an undirected graph with an Eulerian circuit was called an Eulerian graph, and a graph with an Eulerian path but no Eulerian circuit was called a semi-Eulerian graph). For an undirected graph, an Eulerian graph must be connected and have no odd degree vertexes, and a semi-Eulerian graph must be connected and have two odd degree vertexes (one of these two vertexes is the starting point and the other is the ending point). For a digraph, the in-degree and out-degree of each point must be equal to be an Eulerian graph. The Eulerian path of a digraph is the starting point degree-1, the ending point degree is 1, and the rest of the points are 0. The key to a hamilton graph was to traverse all the vertexes without repeating them. If a graph does not meet the criteria of the Eulerian graph, that is, the undirected graph is not connected or has an odd number of degrees,(For a directed graph, the corresponding conditions for in-degree and out-degree are not satisfied), and the requirements for a hamiltonian graph are not satisfied (There are many ways to determine a hamiltonian graph, such as satisfying certain dissimilarity conditions, but there is no simple and unified determination based on the degree of the vertexes like the Eulerian graph. However, the whole graph must be able to traverse all the vertexes without repeating), then this graph is neither an Eulerian graph nor a hamiltonian graph. For example, a graph that is disconnected and has a chaotic structure and cannot traverse all the vertexes is neither an Eulerian graph nor a Hamilonian graph. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
There is a non-trivial solution
In matrix algebra, if the value of the determinate is 0, then there is a non-trivial solution (non-zero solution); if the value of the determinate is not equal to 0, then there is only a trivial solution (X = 0). This concept was part of advanced mathematics. When solving boundary value problems, one might need to consider finding a value that made the problem have a non-trivial solution (non-zero solution). The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
How to distinguish between non-trivial and trivial solutions?
In a system of linear equations, a trivial solution was a solution where all the unknown variables were zero, while a non-trivial solution was a solution where there was at least one non-zero solution. For example, for a linear equation AX = 0, if the coefficient matrix was simplified by the Gauss elimination method to obtain all zero rows, then the unknown variables of the equation could be taken as zero, which was the trivial solution. This meant that all variables in the equation were free variables and there were infinite solutions. If the non-zero rows were obtained after the reduction, there was at least one non-zero solution, which was a non-trivial solution. This meant that there were constraints in the equation and there was more than one solution. In matrix algebra, there were trivial and non-trivial solutions to the zero distribution problem. For example, in Riemann's hypothesis, the trivial solution was all negative even numbers, while the non-trivial solution was more difficult to solve. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!
Non-trivial functional dependence and trivial functional dependence
Let X and Y be the attributes of a relation, and X→Y. If Y is contained in X, then X→Y is called a trivial functional dependence. If Y is not contained in X, then X→Y is called a non-trivial functional dependence. The trivial functional dependence was automatically established because it was determined by the reflexive nature of the functional dependence. The functional dependence that was generally studied was mostly non-trivial functional dependence. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!