To explain why the second neighborhood $N_2(v)$ contains exactly $t(t - 1)$ vertices, let’s break down the problem step by step.

Definitions:

• $G$ is a $t$-regular graph, meaning every vertex has exactly $t$ neighbors.
• The graph has no cycles of length 3 (triangles) or 4.
• $N_1(v)$ refers to the first neighborhood of vertex $v$, i.e., the set of vertices directly connected to $v$.
• $N_2(v)$ refers to the second neighborhood of vertex $v$, i.e., the set of vertices that are connected to $v$ via exactly two edges.

Step-by-Step Explanation:

1. Vertex $v$ and its first neighborhood $N_1(v)$: Since $G$ is $t$-regular, vertex $v$ has exactly $t$ neighbors. Let’s call this set $N_1(v)$. So, $|N_1(v)| = t$.

2. No cycles of length 3 or 4: Since there are no cycles of length 3 (triangles), the neighbors of $v$ in $N_1(v)$ are not connected to each other. This means there are no edges between any two vertices in $N_1(v)$. So, all the neighbors of $v$ lead to new vertices in the second neighborhood.

3. Neighbors of vertices in $N_1(v)$: Each vertex in $N_1(v)$ is also $t$-regular, meaning it has exactly $t$ neighbors. However, one of those neighbors is $v$, which we already accounted for. So, each vertex in $N_1(v)$ has $t - 1$ neighbors that are distinct from $v$.

4. Counting the vertices in the second neighborhood $N_2(v)$: Since the vertices in $N_1(v)$ are not connected to each other (no cycles of length 3 or 4), the $t - 1$ neighbors of each vertex in $N_1(v)$ must all be distinct. Therefore, each vertex in $N_1(v)$ contributes $t - 1$ distinct vertices to the second neighborhood $N_2(v)$.

5. Total number of vertices in $N_2(v)$: There are $t$ vertices in $N_1(v)$, and each of them contributes $t - 1$ distinct vertices to $N_2(v)$. Therefore, the total number of vertices in $N_2(v)$ is: $|N_2(v)| = t(t - 1)$

Conclusion:

The second neighborhood $N_2(v)$ contains exactly $t(t - 1)$ vertices because:

• $v$ has $t$ neighbors, each of which has $t - 1$ distinct neighbors (excluding $v$).
• The absence of cycles of length 3 and 4 ensures that the neighbors of $v$ are not connected to each other, leading to distinct vertices in $N_2(v)$.

Would you like further clarification or details on this? Here are 5 related questions:

1. What is the general definition of a $t$-regular graph?
2. How do the absence of triangles affect graph properties?
3. How would the result change if there were cycles of length 3 or 4?
4. How does the degree of vertices in $N_1(v)$ affect $N_2(v)$?
5. Can this concept be extended to graphs with cycles of longer lengths?

Tip: When counting neighborhoods in a graph, always consider the restrictions imposed by cycles and regularity. These directly impact the number of distinct vertices at different distances from a given vertex.