<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; "><BR><DIV><DIV>On 02/08/2005, at 3:47 AM, Ray Chandler wrote:</DIV><DIV><BR class="khtml-block-placeholder"></DIV><BR class="Apple-interchange-newline"><BLOCKQUOTE type="cite"> <DIV dir="ltr" align="left"><FONT face="Arial" color="#0000ff" size="2"><SPAN class="421071819-01082005">It's not clear to me what differentiates the "multigraph" sequences A000421/A005965 from the "graph" sequences A005638/A002851.</SPAN></FONT></DIV> <DIV dir="ltr" align="left"><FONT face="Arial" color="#0000ff" size="2"><SPAN class="421071819-01082005"></SPAN></FONT> </DIV> </BLOCKQUOTE></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"><BR class="khtml-block-placeholder"></SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"><BR class="khtml-block-placeholder"></SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;">"Graph" means "simple graph" in that any two vertices can be joined by at most one edge, while "multigraph" means "unrestricted graph" in that there can be any number of edges between two vertices - of course each of these edges contributes one to the valency (aka degree) of its endvertices and so for trivalent (aka cubic) graphs there can only be 1, 2 or 3 edges between any two vertices.</SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"><BR class="khtml-block-placeholder"></SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;">Quick example: there is one cubic multigraph on 2 vertices, namely two vertices joined by 3 edges, while there are no cubic graphs on 2 vertices.</SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"><BR class="khtml-block-placeholder"></SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;">All are silently assumed to be loopless, which is an assumption often made by graph theorists, but probably should be stated formally in the definition of these sequences...</SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"><BR class="khtml-block-placeholder"></SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"><BR class="khtml-block-placeholder"></SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;">Cheers</SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"><BR class="khtml-block-placeholder"></SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;">gordon</SPAN></FONT></DIV><DIV style="text-align: -khtml-left;"><FONT class="Apple-style-span" color="#0000FF" face="Arial" size="2"><SPAN class="Apple-style-span" style="font-size: 10px;"></SPAN></FONT></DIV></BODY></HTML>