Supersubdivision in the Context of AUM Block Labeling for Coconut Tree and Double Coconut Tree Graph

Abstract

A graph \(G\) with \(p\) number of vertices, \(q\) number of edges and \(b\) number of blocks, where \(p,q,b\ge 1\). The set of vertices, edges and blocks are denoted by \(V(G)=\{v_1,v_2,\dots ,v_p\}\), \(E(G)=\{e_1,e_2,\dots ,e_q\}\) and \(B(G)=\{b_1,b_2,\dots ,b_n\}\), respectively. The graph \(G\) admits AUM block labeling if there exists a bijection \(f:V(G)\rightarrow z^+\) induced from \(f\) by \(f^*(uv)=f(u)*f(v)\) and \(f^{**}(B_j)=B(G)\rightarrow z^+\) then \(f^{**}(B_j)=\sum^n_{i=1}{f}(v_{ij})+\sum^n_{i=1}f^*(e_{ij})\) and \(f^{**}(B_i)\neq f^{**}(B_j)\). In this paper, we prove that the super subdivision of coconut tree and double coconut tree graph admits AUM block labeling.