Undirected graph

Undirected

Description:

• All edges are undirected

terminology

1. adjacent (neighbors): if 2 endpoints a, b are connected with an edge e
   • a and b are adjacent
   • w is incident with b and c
2. neighborhood
   • $N(f)=\{b,a,e,c\}$
   • $N(a,b)=N(a)\cup N(b)=\{f\}$
3. degree in undirected graph
   • number of edges incident with it, with a loop counted twice to the degree of that vertex
   • $deg(v)$
   • pendant: vertex with degree 1
   • isolate: vertex with degree 0

Handshaking theorem:

• definition: find sum of degrees or total edges of a undirected graph
  • Let $G=(V,E)$ be an undirected graph with 𝑚 edges.
  • Then $2m={\Sigma }_{v\in V}\text{deg}(v)$ (sum of total degrees = 2 x edges)
• Also applies even if multiple edges and loops are present.
• Theorem:
  • An undirected graph has an even number of vertices of odd degreetheorem