Yes, the statement is true.

Any two isomorphic graphs have the same degree sequence. This is because an isomorphism between two graphs preserves the adjacency relationships between the vertices. Consequently, the degree (number of edges connected to a vertex) of each vertex in one graph must match the degree of its corresponding vertex in the other graph.

Explanation:

1. Isomorphic graphs are essentially the same graph represented in two different forms. They can be transformed into one another by relabeling the vertices.
2. Since the degree of a vertex is determined purely by the number of edges it shares with other vertices, the degree sequence remains unchanged under isomorphism.

Important Note:

The converse is not always true: two graphs having the same degree sequence does not necessarily mean they are isomorphic. They could still be different in structure (e.g., non-isomorphic graphs that are degree-sequence equivalent).

Would you like a detailed example or a deeper explanation of this concept?

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Related Questions:

1. What is the formal definition of graph isomorphism?
2. Can two graphs with the same degree sequence be non-isomorphic? Give examples.
3. How do you verify if two graphs are isomorphic?
4. What role does the degree sequence play in graph theory?
5. Can we always determine isomorphism using computational algorithms?

Tip:

When checking for isomorphism, start by comparing simple invariants like the number of vertices, edges, and degree sequences before attempting more complex methods.