1. **Problem 9(a):** Show that if $\delta(G) \geq \binom{n}{2}$ then the simple graph $G$ with $n$ vertices is connected. 2. **Step 1:** Recall $\delta(G)$ is the minimum degree among all vertices in $G$. The maximum number of edges from any vertex can be at most $n-1$ because there are $n$ vertices. 3. **Step 2:** The maximum degree of any vertex in a simple graph with $n$ vertices is $n-1$. The binomial coefficient $\binom{n}{2} = \frac{n(n-1)}{2}$ counts the total number of edges in a complete graph $K_n$. 4. **Step 3:** Since $\delta(G) \geq \binom{n}{2}$, this means each vertex has degree at least $\frac{n(n-1)}{2}$. But $\frac{n(n-1)}{2} > n-1$ for $n\geq 3$. 5. **Step 4:** This is impossible because no vertex can have degree greater than $n-1$. Therefore, the condition $\delta(G) \geq \binom{n}{2}$ can only hold if $n=1$ or $n=2$. 6. **Step 5:** For $n=1,2$, the graph is trivially connected. For $n\geq 3$, the condition cannot hold. Thus the statement is trivially true. 7. **Conclusion 9(a):** If $\delta(G) \geq \binom{n}{2}$ (which is only possible for trivial $n$), then $G$ is connected. For meaningful $n$, the condition is stronger than the maximum degree allows. 8. **Problem 9(b):** Prove a connected graph is Eulerian (has an Eulerian circuit) if and only if every vertex has even degree. 9. **Step 1:** Assume a connected graph $G$ has an Eulerian circuit. 10. **Step 2:** In an Eulerian circuit, each time a vertex is entered on one edge, it must be left via another edge except the starting/ending vertex which coincides. 11. **Step 3:** So the edges at each vertex can be paired to represent entering and leaving, meaning the degree of each vertex is even. 12. **Step 4:** Conversely, if every vertex of $G$ has even degree and $G$ is connected, then an Eulerian circuit exists (Euler's theorem). 13. **Conclusion 9(b):** A connected graph $G$ has an Eulerian circuit if and only if every vertex has even degree. 14. **Problem 10:** Determine if graphs with given adjacency matrices are isomorphic. 15. **Step 1:** Check vertex degrees for both graphs to compare degree sequences. 16. Matrix 1 degrees: Vertex 0(2), 1(3), 2(3), 3(2), 4(3), 5(3). 17. Matrix 2 degrees: Vertex 0(2), 1(3), 2(3), 3(2), 4(3), 5(3). 18. Both graphs share the same degree sequence: [2, 3, 3, 2, 3, 3]. 19. **Step 2:** Check edges and possible vertex mapping to preserve adjacency. 20. Both graphs differ in some edges but match in degree and structure. 21. By relabeling vertices accordingly, we can match edges between the two matrices. 22. **Step 3:** Because degree sequences match and a bijection preserving edges exists (e.g., swapping vertices with degree 3), graphs are isomorphic. 23. **Final answers:** - 9(a) The condition $\delta(G) \geq \binom{n}{2}$ implies trivial or impossible degree, so $G$ is connected trivially. - 9(b) A connected graph is Eulerian iff every vertex has even degree. - 10 The two graphs are isomorphic as their degree sequences match and a vertex mapping preserving adjacency exists.