[graph][example] graph vertix, edge, degree etc

Defination

point  : A particular position which is usually represted by a dot: whether in a one-dimensional, two-dimensional, or three-dimensional space.
line   : The notion of line or straight line between 2 points.
vertex : A vertex is a point where multiple lines meet, it's also so-called node.
edge   : An edge is a particular type of line segment joining two vertices, sometimes we call it path.
adjacent edge   : two different edges have both vertices common.
adjacent vertex : two different verteix have both edges common.
loop   : loop is an edge is drawn from vertex to itself.
degree of a vertex : The degree of a vertex is the number of edges incident with that vertex.
Degree Sequence: Degree sequence of a graph is the list of degree of all the vertices of the graph
indegree  : Indegree of vertex V is the number of edges which are coming into the vertex V.
outdegree : Outdegree is the number of edges which are going out from the vertex V.
isolated vertex  : A vertex without any indegree and outdregree.
parallel edges   : A pair of vertices is connected by more than one edge, these edges are parallel edges.
directed edge    : means the edges have an orientation which is represented by an arrow: the tail and head of this arrow are the nodes representing the beginning and ending points of the edge.
undirected graph : An undirected graph has no directed edges.
directed graph   : An directed graph has directed edges.
Pendant point    : vertex (also leaf vertex) is a vertex with only degree one.

Undirected graph

vertices : {a,b,c,d}
edges    : {ab,ab,bc,cd,bd}
adjacent edges 
 - adjacent edge of vertix "a" are {ab1,ab2}
 - adjacent edge of vertix "b" are {ab1,ab2,cb,bd}
 - adjacent edge of vertix "c" are {cb,cd}
 - adjacent edge of vertix "d" are {bd,bc}
adjacent vertex:
 - adjacent vertex of edge ab1 and ab2 is a.
 - adjacent vertex of edge {ab1,ab2,cd,bd} is b.
 - adjacent vertex of edge bc and bd is c.
 - adjacent vertex of edge cd and bd is d.

(Remark. The figure above is a multi graph.)

 

Vertex    Connecting to      Degree
-------------------------------------
  a          b              deg(a) = 2
  b          a,c,d         deg(b) = 2
  c          c,b           deg(c) = 4
  d          b,c           deg(d) = 2

 

from this table we can found the degree sequence of {a,b,c,d} is {2,4,2,2}

 

 

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Directed graph

vertices                 : {A,B,C,D,E,F,G,H<I,J}
edges                     : {AD,DF,DG,DH,BD,BE,EG,CE,CH}
pandent vertices  : {A,F,J}
isolated vertex     : {I,J}
loop                       : {J}

 

Vertex   in-degree  out-degree
------------------------------
  A          0           1
  B          0           2
  C          0           2
  D          2           3
  E          2            1
  F          1            0
  G         2            0
  H         2            0
  I          0            0
  J          1            1

 

adjacent edges :
 - adjacent edge of vertix B are {BD,BE}
 - adjacent edge of vertix C are {CE,CH}
 - adjacent edge of vertix D are {AD,BD,DF,DG,DH}
 - adjacent edge of vertix E are {BE,EG}
 - adjacent edge of vertix G are {DG,EG}
 - adjacent edge of vertix H are {DH,CH}

 

adjacent vertex:
 - adjacent vertex of edge {AD,BD,DH,DG,DF) is vertix D.
 - adjacent vertex of edge {BE,EG,CE} is E.
 - adjacent vertex of edge {CH,DH} is H.
 - adjacent vertex of edge {DG,EG} is G.

 

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Reference:

http://mathworld.wolfram.com/PendantVertex.html
https://en.wikipedia.org/wiki/Vertex_(graph_theory)
https://www.tutorialspoint.com/graph_theory/graph_theory_fundamentals.htm
https://www.math.cuhk.edu.hk/~wei/ch2.pdf
https://en.wikipedia.org/wiki/Loop_(graph_theory)
https://www.quora.com/When-are-two-edges-said-to-be-adjacent-in-graph-theory
http://rosalind.info/glossary/directed-edge/
http://www.people.vcu.edu/~gasmerom/MAT131/graphs.html