Node? node
I remember Professor Gu of Nankai University once mentioned such an example in an article about mathematics and culture:
"Konigsberg is a beautiful city in Europe. There is a river that runs through the city. There are two small islands in the river, and there are seven bridges between them. When people walk along the river after supper, they often cross the bridge to the island or the other side. One day, someone came up with a game. He suggested that we should not cross the seventh bridge repeatedly and see who can find the route first. This has aroused the interest of many people, but the result of the attempt is that no one can do it. A bridge is lost or a bridge is duplicated.
After many failed attempts, someone wrote to Euler, the great mathematician at that time, for advice. After thinking, Euler first abstracted islands and shores as "points" and bridges as lines. Then Euler abstracted the problem of the Seven Bridges in Konigsberg as "a problem": the pen tip does not leave the paper, and a given figure is drawn stroke by stroke, and no lines are allowed to repeat, which is called "a stroke" for short. The problem to be solved is to find the necessary and sufficient condition of "a figure can draw a stroke" and give the method of drawing a stroke for the figure that can draw a stroke.
After research, Euler successfully solved the above problems, and wrote a paper, which was read at the podium of the Petersburg Academy of Sciences. Euler divides the points on the graph into two categories: note that each point is the endpoint of several lines, and if there are even lines ending at a certain point, it is called an even node; If an odd line ends at a point, it is called an odd node. If you want to draw a figure without repeating a stroke, then if you draw a line at every point except the starting point and the ending point, you must draw a line, which must be an even number of nodes. Therefore, the necessary condition of "one stroke" is "there are no more than two odd nodes in the graph". And vice versa: if there are no more than two odd nodes in the graph, of course, one can be completed. When there are two odd nodes in a graph, one can be used as the starting point and the other as the ending point to complete a stroke. When there are no odd nodes in the graph, a stroke can be completed from any point. There won't be only one odd node in the graph, because every line has two endpoints. In this way, Euler obtained a necessary and sufficient condition that a graph can be drawn: there are no more than two odd nodes in the graph. It can be seen that there are four odd nodes in the graph, so the graph cannot be drawn. No wonder all attempts at the game "Seven Bridges Don't Repeat" failed.
From this example, we deeply feel the powerful power of mathematical abstraction, which also created topology. "
(For the original text, see:/jxyj _ 5312/20060901/t2006090194497.shtml)
From the above understanding, nodes should refer to endpoints.
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