SUBDIVIDED SHELL FLOWER GRAPHS: $\rho$ LABELING

A ρ- labeling (or ρ- valuation) of a graph is an injection from the vertices of the graph with 'q' edges to the set {0,1,2,3,...(2q-1),2q} where if the edge labels induced by the absolute value of the difference of the vertex labels are a1,a2,a₃,...,a_q-1,aq then a_i=i or a_i=(2q+1-i). A shell graph, C(n,n-3), is defined as a cycle C_n with (n - 3) chords sharing a common endpoint called the apex. In other words, a shell graph is the join of complete graph K₁ and P_m, the path with m vertices. A subdivided shell graph is obtained from the shell graph G = P_m ∨ K₁ by subdividing the edges in the path P_m of the shell graph. A subdivided shell flower graph is defined as a one vertex union of k copies of the subdivided shell graph and k copies of the complete graph K₂. In this paper, we prove that subdivided shell ower graphs admit ρ -labeling.