Question 1176885: If a simple, connected, graph has 20 vertices, what is the maximum number of edges it can have? (Recall that a simple graph does not have loops and does not have multiple / parallel edges)
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to determine the maximum number of edges in a simple, connected graph with 20 vertices:
**Understanding the Concept**
* A simple graph means no loops (edges connecting a vertex to itself) and no multiple edges (more than one edge between the same pair of vertices).
* To maximize the number of edges, we want every vertex to be connected to every other vertex. This is called a complete graph.
**Formula for Complete Graphs**
The number of edges in a complete graph with 'n' vertices is given by the formula:
* Edges = n(n - 1) / 2
**Applying the Formula**
In our case, n = 20. So, the maximum number of edges is:
* Edges = 20(20 - 1) / 2
* Edges = 20(19) / 2
* Edges = 380 / 2
* Edges = 190
**Answer**
The maximum number of edges a simple, connected graph with 20 vertices can have is 190.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
If a simple, connected, graph has 20 vertices, what is the maximum number of edges it can have?
(Recall that a simple graph does not have loops and does not have multiple / parallel edges)
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The solution by @CPhill is  ,  , and   .
It is incorrect, because it includes great numbers of loops that are PROHIBITED: triangles,
quadrilateral, pentagons, hexagons, heptagons, octagons, 9-gons, 10-gons, etc. . . . until 20-gon.
The correct solution is mine, which follows below.
DEFINITION:
A simple connected graph is a graph where there is a path between every pair of vertices,
and it does not contain any loops or multiple edges between the same pair of vertices.
In other words, it's a connected graph without any self-loops (edges that start and end
at the same vertex) or multiple edges connecting the same two vertices.
Since there are 20 nodes in our problem, and each node should be connected with at least one of the others,
the minimum number of edges is 19 = 20-1, i.e., 19 connections.
An example of such a graph is
*---*---*---*---*---*---*---*---*---*---*----*---*---*---*---*---*---*---*---* ( <<<---=== 19 edges )
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
But having in any simple, connected graph of 20 nodes more than 19 edges means HAVING a LOOP.
So, the maximum possible number of edges in a 20-nodes simple, connected graph with no loops is  .
Solved.
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I looked at the Google-AI solution to this problem.
Of 03/04/2025, this solution, which is under the link
https://www.google.com/search?q=If+a+simple%2C+connected%2C+graph+has+20+vertices%2C+what+is+the+maximum+number+of+edges+it+can+have%3F+(Recall+that+a+simple+graph+does+not+have+loops+and+does+not+have+multiple+%2F+parallel+edges)&rlz=1C1CHBF_enUS1071US1071&oq=If+a+simple%2C+connected%2C+graph+has+20+vertices%2C+what+is+the+maximum+number+of+edges+it+can+have%3F+(Recall+that+a+simple+graph+does+not+have+loops+and+does+not+have+multiple+%2F+parallel+edges)&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTI4MTZqMGoxNagCCLACAfEFiTVLCcDSTMI&sourceid=chrome&ie=UTF-8
is incorrect, too.
The solution by @CPhill is a copy-paste of that incorrect Google-AI solution.
Surely, I reported to Google AI about their incorrect solution.
Hope it will help them to fix their artificial mind.
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Regarding the post by @CPhill . . .
Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.
The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.
It has no feeling of shame - it is shameless.
This time, again, it made an error.
Although the @CPhill' solution are copy-paste Google AI solutions, there is one essential difference.
Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.
All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.
Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.
And the last my comment.
When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.
Without it, their reliability is ZERO and their creadability is ZERO, too.
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