A
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si5.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=56ae4f9a2679085b42b29a239000f648" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode">-ranking of a directed graph
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si7.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=b61cfbe55c51874ee7b1938dadcc66e9" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode"> is a labeling of the vertex set of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si7.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=b61cfbe55c51874ee7b1938dadcc66e9" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode"> with
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si5.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=56ae4f9a2679085b42b29a239000f648" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode"> positive integers such that every directed path connecting two vertices with the same label includes a vertex with a larger label in between. The
rank number of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si7.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=b61cfbe55c51874ee7b1938dadcc66e9" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode"> is defined to be the smallest
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si5.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=56ae4f9a2679085b42b29a239000f648" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode"> such that
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si7.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=b61cfbe55c51874ee7b1938dadcc66e9" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode"> has a
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0972860015300384&_mathId=si5.gif&_user=111111111&_pii=S0972860015300384&_rdoc=1&_issn=09728600&md5=56ae4f9a2679085b42b29a239000f648" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode">-ranking. We find the largest possible directed graph that can be obtained from a directed path or a directed cycle by attaching new edges to the vertices such that the new graphs have the same rank number as the original graphs. The adjacency matrix of the resulting graph is embedded in the Sierpiński triangle.
We present a connection between the number of edges that can be added to paths and the Stirling numbers of the second kind. These results are generalized to create directed graphs which are unions of directed paths and directed cycles that maintain the rank number of a base graph of a directed path or a directed cycle.