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give me the answer like mathmetical turm not theorytical and all so be with diagram

To solve this problem, we need to apply the Breadth-First Search (BFS) algorithm to find a spanning tree for the given graph. I'll break down the solution in mathematical terms and provide a step-by-step outline of how BFS would build the spanning tree from a selected starting node. 

Given the structure of BFS, we'll start at a specific node (let's choose node \( a \) as the root) and traverse level by level, visiting each neighboring node before moving to the next level.

### Step-by-Step BFS Solution with the Graph

1. **Start at Node \( a \):**
   - **Visited nodes:** \( a \)
   - **Queue:** \( a \)
   
2. **Visit neighbors of \( a \):**
   - **Neighbors of \( a \)** are \( b \), \( e \), and \( d \).
   - Add \( b \), \( e \), and \( d \) to the queue and mark them as visited.
   - **Edges selected:** \( (a, b) \), \( (a, e) \), \( (a, d) \)
   - **Visited nodes:** \( a, b, e, d \)
   - **Queue:** \( b, e, d \)

3. **Visit neighbors of \( b \):**
   - **Neighbors of \( b \)** are \( a \), \( e \), and \( c \).
   - \( a \) and \( e \) are already visited, so we add \( c \) to the queue.
   - **Edge selected:** \( (b, c) \)
   - **Visited nodes:** \( a, b, e, d, c \)
   - **Queue:** \( e, d, c \)

4. **Visit neighbors of \( e \):**
   - **Neighbors of \( e \)** are \( a \), \( b \), \( c \), and \( d \).
   - All neighbors \( a \), \( b \), \( c \), and \( d \) are already visited, so no new nodes are added.
   - **Queue:** \( d, c \)

5. **Visit neighbors of \( d \):**
   - **Neighbors of \( d \)** are \( a \) and \( e \).
   - Both neighbors \( a \) and \( e \) are already visited, so no new nodes are added.
   - **Queue:** \( c \)

6. **Visit neighbors of \( c \):**
   - **Neighbors of \( c \)** are \( b \) and \( e \).
   - Both neighbors \( b \) and \( e \) are already visited, so no new nodes are added.
   - **Queue:** (empty)

At this point, all nodes have been visited, and the edges selected form a spanning tree.

### Resulting Spanning Tree

The edges in the spanning tree obtained through BFS from node \( a \) are:
\[
(a, b), (a, e), (a, d), (b, c)
\]

### Spanning Tree Diagram

The spanning tree structure can be drawn as follows (edges used are highlighted):

- **Node \( a \)** connects to **\( b \)**, **\( e \)**, and **\( d \)**.
- **Node \( b \)** connects to **\( c \)**.

This forms a connected, acyclic graph that spans all vertices, meeting the requirements of a spanning tree.

---

Would you like a visual representation of this spanning tree?

### Related Questions:
1. How would the BFS traversal change if we started from a different node, like \( b \)?
2. What are the key differences between BFS and Depth-First Search (DFS) in terms of resulting spanning trees?
3. How does BFS guarantee finding the shortest path in an unweighted graph?
4. Can BFS be applied to weighted graphs, and what modifications are necessary?
5. How can we verify if the edges obtained indeed form a tree?

### Tip:
In BFS, always process each level of neighbors before moving to the next to ensure the shortest-path characteristic in an unweighted graph.

Write the Breadth-First search (BFS) algorithm and use BFS algorithm to find a spanning tree of the graph.

Learn how to apply the Breadth-First Search (BFS) algorithm to find a spanning tree in a graph. This guide provides a step-by-step mathematical solution, suitable for undergraduate students studying graph theory.

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