This is a Discrete Math question.

Please draw a graph and look through (1)-(10).

Thanks for your help.

Sketch a graph which has:

• Nodes {1,2,3,4,5,6}
• Arcs {a₁, a₂, a₃, a₄, a₅, a₆,a₇ }
• Function: g(a₁) = 1-2, g(a₂) = 1-3, g(a₃) = 3-4, g(a₄) = 3-4

g(a₅) = 4-5, g(a₆) = 5-5, g(a₇) = 6-6

(1) Give two nodes that are adjacent.

(2) Give two nodes that are not adjacent.

(3) Give a node that is adjacent to itself.

(4) Give a loop.

(5) Is the graph simple? If the graph is not simple, give a pair of parallel arcs.

(6) Give a node of degree 2.

(7) Is the graph complete?

(8) Is the graph connected? If the graph is not connected, list the isolated node.

(9) Give a path of length 4 from 2 to 5.

(10) Are there any cycles in the graph?

Reference:

Graph Terminology:

Adjacent: Two nodes are adjacent if they are endpoints associated with an arc.

Loop: A loop is an arc with endpoint n-n for some node n.

Parallel Arcs: Two arcs with the same endpoints.

Simple Graph: A graph with no loops or parallel arcs.

Isolated Node: a node which is not adjacent to any other node.

Degree of a Node: Number of arcs ending at the node.

Complete Graph: A graph in which any two distinct nodes are adjacent.

Path: A path from node n0 to node nk is a sequence n0a0n1a1 ... ak-1ak-1nk that goes from node n0 to node nk.

Length of a Path - the number of arcs in a path.

Connected - A graph is connected if there is a pth from any node to any other node

Cyle - a path from n0 back to n0 where no arcs or nodes are repeated.