Total H-irregularity strength of ladder graphs

Abstract

Abstract H-irregular total labeling and total H-irregularity strength of graphs were not complete.This research aimed to determine the total H-Irregularity strength of ladder graph Lb with subgraphs C 8 and C 10.To determine of the total H-irregularity strength of ladder graphs, we have to determine the greatest lower bound and the smallest upper bound. The lower bound was analysed based on graph characteristics and other supporting theorems, while the upper bound was analysed by construct a function H-Irregular total labeling. The results reveal that the total H-Irregularity strength of ladder graph Ln with subgraph C 8 is <?CDATA $\lceil \frac{n-2i+14}{16}\rceil \le ths({L}_{n},{C}_{8})\le \lceil \frac{n+14}{18}\rceil $?> ⌈ n − 2 i + 14 16 ⌉ ≤ t h s ( L n , C 8 ) ≤ ⌈ n + 14 18 ⌉ , for <?CDATA $18i-15\le n\le 18i+2$?> 18 i − 15 ≤ n ≤ 18 i + 2 , where i = 1,2, …, N and the total H-Irregularity strength of ladder graph Ln with subgraph C 10 is <?CDATA $\begin{array}{c}ths(G,H)\ge \max \{\lceil 1+\frac{{m}_{1}-1}{|V(H/{S}_{1})|+|E(H/{S}_{1})|}\rceil, \ldots, \lceil 1+\frac{{m}_{1}-1}{|V(H/{S}_{z})|+|E(H/{S}_{z})|}\rceil \}\\ \lceil \frac{n-3i+18}{20}\rceil \le ths({L}_{n},{C}_{10})\le \lceil \frac{n+18}{23}\rceil \end{array}$?> t h s ( G , H ) ≥ max { ⌈ 1 + m 1 − 1 | V ( H / S 1 ) | + | E ( H / S 1 )

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