Here are the answers to the questions on your worksheet:

[1-8] Questions:

1. The types of graphs shown are:

   • Graph 1: Complete graph $K_4$ (since each vertex is connected to every other vertex).
   • Graph 2: Multigraph (contains multiple edges between the same pair of vertices).

2. A graph with multiple edges connecting the same pair of vertices is called a multigraph.

3. The set of vertices adjacent (or neighboring) to a vertex $a$ is called the neighborhood of $a$.

4. In the context of graph $G = (V, E)$, where $a \in V$, the set of vertices adjacent to $a$ is often denoted by $N(a)$ or neighborhood of $a$.

5. A vertex with a degree of 0 (no edges connected to it) is called an isolated vertex.

6. A vertex with a degree of 1 is called a pendant vertex or leaf.

7. A graph containing directed edges is called a directed graph or digraph.

8. A subgraph of $G$ that is not identical to $G$ itself is called a proper subgraph.

[9-15] Questions:

9. In a simple, undirected graph with 10 vertices and 10 edges, the number of edges should be 5 if it’s a connected graph (matching up to each vertex’s degree).

10. The last graph in the image can be determined to be bipartite if it can be divided into two disjoint sets such that no two vertices within the same set are adjacent. Answer: O (Yes, it is bipartite).

If you have further questions about any particular answer, let me know.

Additional Questions to Explore:

1. What makes a graph complete, and how does it differ from a multigraph?
2. Can a multigraph have directed edges, or is it limited to undirected edges?
3. How does the neighborhood of a vertex help in analyzing graph properties?
4. What are practical examples of isolated and pendant vertices in real-world networks?
5. Why are bipartite graphs significant, and in what scenarios are they useful?

Tip:

When analyzing a graph, checking for properties like connectivity, degree sequence, and the presence of loops or multiple edges can help quickly classify it into categories such as simple graphs, multigraphs, or bipartite graphs.