1. Graph theory is a branch of mathematics that studies graphs, which are mathematical structures used to model pairwise relations between objects. 2. A graph consists of vertices (also called nodes) and edges (connections between nodes). 3. There are several types of graphs: - **Undirected Graphs:** Edges have no direction. Example: friendship networks. - **Directed Graphs (Digraphs):** Edges have a direction. Example: Twitter follower relationships. - **Weighted Graphs:** Edges carry weights representing costs or distances. Example: road networks with distances. - **Bipartite Graphs:** Vertices can be divided into two disjoint sets with edges only between sets. Example: job assignments. 4. Common models include: - **Complete Graph:** Every pair of distinct vertices is connected by an edge. - **Cycle Graph:** Vertices connected in a closed chain. - **Tree:** A connected graph with no cycles. 5. Graphical representation involves drawing vertices as points and edges as lines or arrows between points. 6. Example: A simple undirected graph with vertices $V=\{A,B,C\}$ and edges $E=\{(A,B),(B,C)\}$ can be drawn with points labeled A, B, C and lines connecting A to B and B to C. 7. These models help solve problems in computer science, biology, social sciences, and more by representing complex relationships visually and mathematically.