"Discrete Mathematics"--Graph theory 6.8

Short line, Distance: The memory gamma is a drop path from VI to VI, γi = min{γ1, gamma 2, Γ3 ...}, then γi is the short line between VI and VJ, the length of which is the distance between the VI and VJ.

Point cut set, cut point: Remember P (g) is the connectivity of G, for the subset V of the point set V ', V ' arbitrary subset V ', satisfies P (g-v ') >p (G), then v becomes a point cut set, when a point cut set only a point V, V is called the cut point.

Edge cut set, cutting Edge (bridge): Analogy point cut set, cut point has a similar concept.

Point connectivity: denoted by κ (g), K (g) = min{| V ' | | V ' is a point cut set or make g-v ' is having a vertex}

Edge connectivity: A similar definition of point connectivity, expressed in λ (G).

From the definition, we can easily get the following knowledge points:

(1): For trivial graphs (only one vertex), κ (g) =λ (g) = 0;

(2): For full graph kn, there is no point cut set, so κ (G) = n-1. (This is also why the definition of point connectivity is more than one or the case)

(3) If there is a cut point in G, κ (g) = 1, if there is a cut edge, λ (g) = 1;

for κ (g), λ (g), Δ (g), the inequality is established (here for the proof of folding).

κ (g) ≤λ (g) ≤δ (g)

The connectivity of the undirected graphs: weakly connected graphs, unidirectional connected graphs, and strong connected graphs.

Weakly connected graph: The undirected graph obtained by removing the directivity of the directed graph, if it is a connected graph, is called a weakly connected graph.

Unidirectional connectivity diagram: A path with at least one point to another point to any two point in the graph.

Strongly connected graphs: any of the two points in the direction graph can reach each other.

It can be seen that the connectivity of weakly connected graphs, unidirectional connected graphs, and strong connected graphs is enhanced in turn, that is, strong connected graphs must be unidirectional connected graphs, one-way connected graphs must be weak connected graphs, and vice versa.

Here are two simple ways to judge the connectivity of a graph.

Discriminant method One: If there is a loop that traverses all the vertices in the graph G, it can be judged to be a strongly connected graph.

Discriminant method Two: If there is a way to traverse all vertices in the graph G, it can be judged as a weakly connected graph.

The correctness of this is self-evident, but we should note that these two conditions of judgment appear as sufficient and not necessary.

Matrix Representations of graphs:

First of all, the matrix of several representations, it can be divided into two categories, one is to characterize the vertex and Edge Association of the number of times, such as correlation matrix, the other is to characterize the relationship between vertices and vertices, such as the adjacency matrix and the matrix, the difference is the former two points between the number of direct access, The latter is to record whether there is a direct path between two points.

The correlation matrix of the graph: the graph G has n vertex m edges, then the n row M-column matrix M,mij represents the number of times the VI and EJ are associated (i.e. 0, 1, 2).

Associative matrices with a direction-free graph:

Similar to the non-graph association matrix, but here the value of the matrix is slightly different, for the Mij value, like the following three kinds of situations:

(1) VI is not associated with EJ, mij = 0;

(2) VI is the starting point of EI, mij = 1;

(3) VI is the focus of EI, mij=-1;

(4)

Adjacency matrix of a directed graph: for AIJ, the number of edges that represent VI->VJ (direct connectivity)

There is a very important theorem about the adjacency matrix of a graph:

Set A is the adjacency matrix of directed graph D, the fixed point of D and v={v1,v2,v3...vn}, then the element aij of A^l (l≥1) is the number of paths of VI to VJ length L, which is the number of paths in D which is length L in D.

Proof: Consider using mathematical inductive method.

Inductive basis: when m = 1, there is the definition of adjacency matrix, the conclusion is obviously established.

Induction step: Assuming that when m = L–1 the theorem we give is set up, then we now consider getting the path of length l, we need to find all the points VK to meet <vi,vk> = l, <vk,vj> = 1, which can be expressed in the following equation:

It can be seen that this is exactly the same as the principle of matrix operation, then based on m=l-1, M = L is also set up, and thus the completion of the theorem is concluded.

The reach matrix of the graph: for Pij, meet (vi can reach VJ)? (Pij = 1): (Pij = 0).

"Discrete Mathematics"--Graph theory 6.8