| Summary: | 碩士 === 國立暨南國際大學 === 資訊工程學系 === 98 === For a graph G = (V;E), a vertex (edge, respectively) t-ranking is a coloring c : V !
f1; 2; : : : ; tg (c0 : E ! f1; 2; : : : ; tg, respectively) such that, for any two vertices u and
v (edges ex and ey, respectively) with c(u) = c(v) (c0(ex) = c0(ey), respectively), every
path between them contains an intermediate vertex w (edge ew, respectively) with
c(w) > c(u) (c0(ew) > c0(ex), respectively). The vertex ranking number r(G) (edge
ranking number 0
r(G), respectively) is the smallest value of t such that G has a vertex
(edge, respectively) t-ranking. The problem to nd r(G) (0
r(G), respectively) for a
graph G is called the vertex ranking problem (the edge ranking problem, respectively).
A partition of V is a set of nonempty subsets of V such that every vertex in V is in
exactly one of these subsets. In this thesis, we introduce two relations for ranking
number of a graph. One is between vertex ranking number and vertex partitions, the
other is between edge ranking number and vertex partitions. By using the proposed
recurrence formulas, we derived two results. One is the edge ranking number of
the Sierpinski graph 0
r(S(n; k)) = n0
r(Kk) for any n; k > 2, and the other is the
bound of vertex ranking number of Sierpinski graphs (n2)
r(Kk)+r(S00(n; k)) 6
r(S(n; k)) 6 (n2)0
r(Kk)+r(S00(n; k))+1 for any n > 2 and k > 3 where S00(n; k)
is the subgraph of S(n; k) by removing all vertices of degree k 1 from S(n; k).
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