To show that if $G$ is a non-regular graph of order $n$ and size $\frac{rn}{2}$ for some integer $r$ with $1 \leq r \leq n - 2$, then the difference between the maximum degree $\Delta(G)$ and the minimum degree $\delta(G)$ is at least 2, we can proceed as follows:

Definitions:

1. Order $n$: The number of vertices in the graph $G$.
2. Size $m$: The number of edges in the graph $G$. Here, $m = \frac{rn}{2}$.
3. Maximum degree $\Delta(G)$: The highest degree of any vertex in $G$.
4. Minimum degree $\delta(G)$: The lowest degree of any vertex in $G$.
5. Non-regular graph: A graph where not all vertices have the same degree.

Proof:

1. Graph parameters:

   • $G$ has $n$ vertices.
   • $G$ has $\frac{rn}{2}$ edges.

2. Degree sum formula:

   • The sum of the degrees of all vertices in $G$ is $2m$ (since each edge contributes to the degree of two vertices).
   • Thus, $\sum_{v \in V(G)} \deg(v) = 2m = rn$.

3. Average degree:

   • The average degree of a vertex in $G$ is $\frac{\sum_{v \in V(G)} \deg(v)}{n} = \frac{rn}{n} = r$.

4. Non-regularity condition:

   • $G$ is non-regular, which means there exists at least one vertex with a degree higher than $r$ and at least one vertex with a degree lower than $r$.
   • Let $\Delta(G)$ be the maximum degree of $G$ and $\delta(G)$ be the minimum degree of $G$.

5. Degree constraints:

   • Since $G$ is non-regular, we have $\Delta(G) > r$ and $\delta(G) < r$.

6. Establishing the bound:

   • Consider the extreme cases:
     • If $\Delta(G) = r+1$ and $\delta(G) = r-1$: $\Delta(G) - \delta(G) = (r+1) - (r-1) = 2.$
   • This is the smallest difference between the maximum and minimum degrees in a non-regular graph that satisfies the given constraints.
   • If $\Delta(G)$ were greater than $r+1$ or $\delta(G)$ were less than $r-1$, the difference $\Delta(G) - \delta(G)$ would be even larger.

7. Conclusion:

   • Therefore, the minimum possible value for $\Delta(G) - \delta(G)$ in a non-regular graph of order $n$ and size $\frac{rn}{2}$ is 2.
   • Hence, $\Delta(G) - \delta(G) \geq 2$.

Would you like more details or have any questions?

Here are five numbered questions you might want to ask next:

1. Can you provide an example of a non-regular graph that satisfies these conditions?
2. How do these properties change if $G$ were a regular graph?
3. What are the implications of this result in graph theory?
4. Can you explain more about the degree sum formula?
5. How do we generalize this result to other types of graphs?

Tip: When proving inequalities involving graph parameters, it's often helpful to consider the definitions and properties of the graph, and to think about the extreme cases that could establish the bounds.