document.write( "Question 3797: Every man on earth has made a certain number of handshakes.Prove that the number of people who have made an odd number of handshakes is even. \n" ); document.write( "
Algebra.Com's Answer #1660 by khwang(438)\"\" \"About 
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This is a basic fact of graph theory.
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\n" ); document.write( " Every person (not every man !) as a vertex and a handshakes between two
\n" ); document.write( " peresons mean the two vertices connecting by an edge in the graph. In
\n" ); document.write( " other word, for each edge (handshake) corresponding to two person.)\r
\n" ); document.write( "\n" ); document.write( " For each vertex v(person), the number of vertices adjacent to it is
\n" ); document.write( " called the degree (# of his/her handshakes).
\n" ); document.write( " For any graph G,(ieforany group of people), the summation of deg(v)
\n" ); document.write( " for all vertex in G must be equal to 2|E| (where |E| is the number
\n" ); document.write( " of edges in G, ie # of handshakes.)
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\n" ); document.write( " Next among all persons (all vertices in the graph G), some has
\n" ); document.write( " odd deg while others are of even degree. But, the # of vertices
\n" ); document.write( " with odd degree (odd # of handshakes) should be even. For otherwise,
\n" ); document.write( " the total degree of the odd vertices would be odd (why ?). Thus,
\n" ); document.write( " adding the even vertices would cause the total degree becoming odd.
\n" ); document.write( " This contradicts to the previous claim.
\n" ); document.write( " Therefore, the number of people who have made an odd number of handshakes is
\n" ); document.write( " even.
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\n" ); document.write( " Further questions are welcome.\r
\n" ); document.write( "\n" ); document.write( " Kenny
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