Questions

No, not every subgraph of a regular graph is regular. A regular graph is one in which every vertex has the same degree. However, when you create a subgraph, you may choose a subset of the vertices and edges from the original graph. During this process, some vertices in the subgraph might have fewer edges than in the original graph, resulting in uneven degrees across vertices in the subgraph.

For example, consider a simple 3-regular graph (each vertex has degree 3) like a triangle with an extra vertex connected to each of the triangle's vertices. If you take a subgraph consisting of just the triangle, it remains regular, but if you take a subgraph by removing only one vertex from the original, the resulting subgraph will not have equal degree for all remaining vertices, hence not regular.

Therefore, a subgraph of a regular graph is not necessarily regular, as subgraphs may lack the uniform degree distribution present in the original graph.