document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176885>Question 1176885</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850556 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850556\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "If a simple, connected, graph has 20 vertices, what is the maximum number of edges it can have? <BR>\n" );
document.write( "(Recall that a simple graph does not have loops and does not have multiple / parallel edges)<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by &nbsp;@CPhill is &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28FATALLY%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28FATALLY%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28TOTALLY%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28TOTALLY%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28GLOBALLY%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28GLOBALLY%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28WRONG%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28WRONG%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is incorrect, &nbsp;because it includes great numbers of loops that are &nbsp;PROHIBITED: &nbsp;triangles, \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;quadrilateral, &nbsp;pentagons, &nbsp;hexagons, &nbsp;heptagons, &nbsp;octagons, &nbsp;9-gons, &nbsp;10-gons, &nbsp;etc. . . . until &nbsp;20-gon.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The correct solution is mine, &nbsp;which follows below.\r<BR>\n" );
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document.write( "DEFINITION:\r\n" );
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document.write( "    A simple connected graph is a graph where there is a path between every pair of vertices, \r\n" );
document.write( "    and it does not contain any loops or multiple edges between the same pair of vertices. \r\n" );
document.write( "    In other words, it's a connected graph without any self-loops (edges that start and end \r\n" );
document.write( "    at the same vertex) or multiple edges connecting the same two vertices. \r\n" );
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document.write( "Since there are 20 nodes in our problem, and each node should be connected with at least one of the others, \r\n" );
document.write( "the minimum number of edges is  19 = 20-1,  i.e., 19 connections.\r\n" );
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document.write( "An example of such a graph is\r\n" );
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document.write( "    *---*---*---*---*---*---*---*---*---*---*----*---*---*---*---*---*---*---*---*      ( <<<---=== 19 edges )\r\n" );
document.write( "    1   2   3   4   5   6   7   8   9   10   11  12  13  14  15  16  17  18  19  20\r\n" );
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document.write( "But having in any simple, connected graph of 20 nodes more than 19 edges means HAVING a LOOP.\r\n" );
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document.write( "So, the maximum possible number of edges in a 20-nodes simple, connected graph with no loops is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%2819%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%2819%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "I looked at the Google-AI solution to this problem.\r<BR>\n" );
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document.write( "Of 03/04/2025, this solution, which is under the link\r<BR>\n" );
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document.write( "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\r<BR>\n" );
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document.write( "is incorrect, too.\r<BR>\n" );
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document.write( "The solution by @CPhill is a copy-paste of that incorrect Google-AI solution.\r<BR>\n" );
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document.write( "Surely, I reported to Google AI about their incorrect solution.\r<BR>\n" );
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document.write( "Hope it will help them to fix their artificial mind.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . \r<BR>\n" );
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document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r<BR>\n" );
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document.write( "The artificial intelligence is like a baby now. It is in the experimental stage <BR>\n" );
document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.\r<BR>\n" );
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document.write( "This time, again, &nbsp;it made an error.\r<BR>\n" );
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document.write( "Although the @CPhill' solution are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.\r<BR>\n" );
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document.write( "Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental<BR>\n" );
document.write( "and can make errors/mistakes.\r<BR>\n" );
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document.write( "All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:<BR>\n" );
document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.<BR>\n" );
document.write( "So, he NEVER SAYS TRUTH.\r<BR>\n" );
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document.write( "Every time, &nbsp;@CPhill embarrassed to tell the truth.\r<BR>\n" );
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document.write( "But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.\r<BR>\n" );
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document.write( "And the last my comment.\r<BR>\n" );
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document.write( "When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, <BR>\n" );
document.write( "until the specialists and experts will check and confirm their correctness.\r<BR>\n" );
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document.write( "Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.\r<BR>\n" );
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