To show that if two graphs $G$ and $H$ are isomorphic, then each pair of corresponding vertices has the same degree, we can proceed as follows:

Definitions

1. Graph Isomorphism: Two graphs $G$ and $H$ are isomorphic if there exists a bijection $f: V(G) \to V(H)$ such that:

   • For every edge $(u, v) \in E(G)$, the edge $(f(u), f(v)) \in E(H)$.
   • Conversely, for every edge $(x, y) \in E(H)$, the edge $(f^{-1}(x), f^{-1}(y)) \in E(G)$.

2. Degree of a Vertex: The degree of a vertex $v$ in a graph is the number of edges incident to $v$.

Proof

1. Let $G$ and $H$ be isomorphic graphs with a bijective mapping $f: V(G) \to V(H)$.

2. Take any vertex $u \in V(G)$. By definition of isomorphism, the image of $u$ under $f$ is $f(u) \in V(H)$.

3. The degree of vertex $u$ in $G$, denoted as $\deg_G(u)$, is the number of edges incident to $u$. Let's denote the edges incident to $u$ as $(u, v_1), (u, v_2), \ldots, (u, v_k)$, where $v_1, v_2, \ldots, v_k$ are the vertices adjacent to $u$.

4. Since $G$ is isomorphic to $H$, each edge $(u, v_i)$ corresponds to an edge $(f(u), f(v_i))$ in $H$. Thus, $f(v_1), f(v_2), \ldots, f(v_k)$ are the vertices adjacent to $f(u)$ in $H$.

5. Therefore, the degree of the vertex $f(u)$ in $H$ is given by: $\deg_H(f(u)) = \text{Number of edges incident to } f(u) = k = \deg_G(u)$

6. This reasoning holds for any vertex $u \in V(G)$. Thus, we conclude that: $\deg_H(f(u)) = \deg_G(u)$ for every vertex $u$.

7. Since $f$ is a bijection, we can say that for any corresponding vertices $u$ in $G$ and $f(u)$ in $H$, they have equal degrees: $\deg_G(u) = \deg_H(f(u))$

Conclusion

Hence, if $G$ and $H$ are isomorphic graphs, then each pair of corresponding vertices of $G$ and $H$ has the same degree.

Questions or Clarifications?

Would you like to explore more about graph isomorphism, or do you have any questions about this proof? Here are some related topics you might find interesting:

1. What are examples of isomorphic graphs?
2. How can we determine if two graphs are isomorphic?
3. What are the implications of isomorphism in graph theory?
4. Can isomorphic graphs have different representations?
5. What are other properties preserved under graph isomorphism?

Tip: To understand graph theory better, try visualizing graphs and their isomorphisms by drawing them out!