Loop path

Yes n A graph of vertices becomes n Order graph , A graph without edges is called a zero graph .

1 A zero order graph is called a trivial graph , A trivial graph has only vertices , There is no side , set up e=（vi,vj） said vi,vj yes e The endpoint of ,e And vi vj relation If vi!=vj That is, there is no self ring , said vi,vj And e The number of associations is in , If vi==vj said vi And e Number of associations by 2.

Simple picture ： There are no duplicate edges or parallel edges .

Subgraph symbol G[ V ] It refers to the point derived subgraph ,G[E] It refers to the edge derived subgraph

Regulations ： Oneself and oneself must be connected

Primary pathway ： stay n From a vertex in a graph u To v Existence path , From u To v The length of existence is n-1 The primary pathway .

Connected graph ： Any two points are connected , A trivial graph is a connected graph .

Connecting branches ：V About R The equivalent class derived subgraph of is G（V1）、G（V2）、、、、 be called G China Unicom Branch , The number of each fulcrum connected is recorded as p（G）, If G It's a connected graph , be p（G）=1, if p（G）>=2, be G It must be a non connectivity diagram .

If all edges in the path are different, the graph is called a simple path , On this basis, if the starting point and the end point coincide , It is called a loop ,

Bribery is a special channel .

If all points in the path are different, all sides are different , The path is called a simple path or path , On this basis , If the starting point and the ending point coincide , It is called the primary loop or loop , A circle with an odd length is called an odd circle , Even numbers are called even circles , If there is repetition in the path, it will appear , It is called complex pathway .

u and v The shortest path between is called geodesic path ,d（u,v） u And v The length of the geodesic path between .

Nature if u And v Disconnected , be d （u,v）= infinity .

How to compare connectivity ： So that the connectivity of a connected graph is destroyed （ Become the second Unicom ） The more edges you need to break , The worse the connectivity of the graph .

G-v from G Delete in v And its associated edges .

G-V‘ from G Delete in V’ All vertices and their associated edges .

G-e from G Delete edges from e .

G-E‘ from G Delete in E’ All the edges in .

In order to enhance the connectivity of graphs , We can choose to delete some vertices and some edges .

Point cut sets ： If we can delete at least a few points to make the graph unconnected , Then the set composed of these points is called point cut set , Note that the cut set is not unique , But at a certain point cut set , The number of elements is the least .

Edge cut sets ： If we can delete at least a few entries, the graph will become disconnected , Then the set composed of these edges is called edge cut set , Note that the edge cut set is not unique , But in a definite edge cut set , The number of elements is the least .

Bridge ： The only path connecting a connected graph .

nature K（n） No point cut set ;

n Zero order graph No point cut set, no edge cut set .

if G Unicom ,E‘ Is an edge cut set , be p3（G-E’）=2;

if G Unicom ,V‘ Is a point cut set , be p（G-V’）>=2;

Point connectivity K（G） Connectivity with edge LEM he （G） The minimum number of elements in the set of cut points , The number of elements in the edge cut set with the smallest element .

Can be up to from u To v There is a way said u Can be up to v, If at the same time v Can be up to u said u And v Reach each other .

D Strong China Unicom ： about arbitrarily u,v ,u And v Reach each other .

D One way connection ： about arbitrarily u,v ,u Can be up to v or v Can be up to u.

D Weak connectivity ： If you omit D The undirected graph obtained by the direction of each edge in is a connected graph , It is called weakly connected graph or connected graph

Incidence matrix of undirected graph ： Make mij For the top vi With the edge ej Number of associations , call （mij）n*m by G The incidence matrix of is recorded as M（G）.

mij There are three values of Namely It's not relevant ,1 The number of associations is 1,2 The number of associations is 2 namely ej In order to vi A ring that is an endpoint .

Incidence matrix of directed acyclic graph ：

With directed acyclic graph D <V,E>, Make mij The table is the degree of correlation ,mij =1, vi by e j Start edge of （ The degree of ）,0 vi And ej It's not relevant ,mij=-1,vi yes ej The end （ The degree of ）, said （mij ）n*m by D The incidence matrix of is recorded as M （D）;

Adjacency matrix of digraph ：

With directed acyclic graph D <V,E> Make aij It means a vertex vi Connect to vertex vj The number of sides , said （aij）n*m by D The adjacency matrix of , Write it down as A（D） Short for A;

Adjacency matrix of undirected graph , Let undirected simple graph G =<V,E>, Make aij For the top vi To the top vj The number of edges between , said （aij）n*m by G The adjacent matrix of is written as A（G）.