## 7.Graphs and Trees

Graphs are used widely to model problems in many different application areas.

• Graph of major highway arteries in western Canada
• Graphical representation of a city street system.
• Graphical representation of a UNIX file system tree
• Graph representation of a computer network
• Graph representation of a software system
• Flow graph notation for various constructs

Definition 7.1 (Graph)

A graph G = (V, E, f) consists of a nonempty set V called nodes of the graph, E is said to be the set of edges of the graph and f is a mapping from the edges E to a set of ordered or unordered pairs of elements of V. If an edge is mapped to an ordered pair, it is called a directed edge; otherwise it is called an undirected edge.

Definition 7.2 (Weighted Graph)

A graph in which weights are assigned to every edge is called a weighted graph.

Definition 7.3 (Isolated Node)

In a graph a node that is not adjacent to any other node is called an isolated node.

Definition 7.4 (Digraph)

A graph G in which every edge is directed is called a digraph, or directed graph.

Definition 7.5 (Outdegree, indegree)

In a digraph, for any node v the number of edges that have v as their initial node is called the outdegree of v.

The number of nodes that have v as their terminal node is called the indegree of v.

The sum of outdegree and indegree of v is called its total degree.

Definition 7.6 (Subgraph)

Let V(H) be the set of nodes of a graph H and V(G) the set of nodes of a graph G such that V(H) V(G). If, in addition, every edge of H is also an edge of G then the Graph H is called a subgraph of the graph G, which is expressed by writing H G.