equivalent graphs
graphs that are the same [all of the same pairs of vertices are adjacent]
Equivalent graphs are two or more graphs that have the same mathematical properties. In other words, they represent the same relationships between various elements or objects. For example, two graphs may have the same number of vertices, edges, and vertices degrees, and they may also have the same number of cycles or connected components.
There are different types of equivalent graphs, including isomorphic graphs and homeomorphic graphs.
Isomorphic graphs are graphs that have the same structure, meaning they have the same number of vertices and edges, and the same edges connect the same vertices in the same way. In other words, they are identical in terms of their topology or shape, but they may have different vertex or edge labels. An example of isomorphic graphs is the complete graph K3 and the cycle graph C3, which both have three vertices and three edges.
Homeomorphic graphs are graphs that can be transformed into each other by a series of graph operations, such as edge contractions and edge deletions. These operations preserve the number of vertices and edges, but they may change the graph structure or connectivity. For example, the graphs K5 and K3,3 are homeomorphic because K5 can be transformed into K3,3 by deleting a single edge, and K3,3 can be transformed into K5 by adding a new vertex and connecting it to all the vertices of one partition.
In summary, equivalent graphs are useful in many mathematical and computational applications, such as graph theory, network analysis, and algorithm design, where it is important to recognize and exploit similarities or symmetries between graphs to solve problems efficiently.
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