Isomorphism is a type of mapping defined between mathematical objects, which can reveal the relationships existing between the attributes or operations of these objects. If there is an isomorphic mapping between these two mathematical structures, then the two structures are said to be isomorphic. Generally speaking, if you ignore the specific definition of the properties or operations of isomorphic objects, structurally speaking, isomorphic objects are completely equivalent-Wikipedia
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1. Simple isomorphism example
Regarding the isomorphic of graphs (Isomorphic), the simplest examples are pentagons and five-pointed stars:
注意：这里的abcde代表的是点，而e1,e1,r3,e4,e5代表的是边
In the above figure, G1 and G2 are isomorphic, because:
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From the node of G1 to the node of G2, there is a one-to-one mapping function f()
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From the edge of G1 to the edge of G2, there is a one-to-one mapping function g()
In G1, edge e1 is associated with nodes a and b, and if and only if (if and only if) in G2, edge g(e) is associated with nodes f(a) and f(b) (E1 and node A, B are related). If this condition is met, the functions f and g are called isomorphism from G1 to G2 (Isomorphism)
PS: The mapping relationship here is as follows: the
lowercase pairs of dots and edges correspond to their uppercase counterparts.
2. Examples of complex isomorphic graphs
The above example is relatively simple and can be seen at a glance, let’s look at an example provided by Wikipedia
For convenience, the corresponding nodes in the two figures are dyed the same color. One thing to note is that in graph theory, a graph can often be drawn on paper or screen in many different ways, so two A graph that looks very different may also be isomorphic . Especially when the number of nodes in the graph is relatively large, it is difficult to judge whether they are isomorphic from the drawn graph at a glance.
3. A more vivid understanding
Briefly summarize the two "sides" of understanding graph isomorphism:
- For a graph, we can think of it as some small balls tied together by ropes. The small balls are the vertices and the ropes are the edges. Now move the ball at will, the ball can go anywhere, and the rope will move around with the ball. The graphs formed at every moment in the movement process are isomorphic.
- Assume that each vertex in the graph has a name, such as: 1, 2, 3,… ,n. Now erase the names on these vertices. After wiping, randomly write new names to these vertices, which can be any name, which is equivalent to changing their "identity". The two figures before and after are isomorphic.
参考链接：
1、https://zh.wikipedia.org/wiki/%E5%9B%BE%E5%90%8C%E6%9E%84
2、https://www.zhihu.com/question/326620873/answer/1063169941
3、https://www.jianshu.com/p/c33b5d1b4cd9