Edge Irregular Reflexive Labeling of Ladder Graph Corona Null Graph Families

Abstract

Given a graph $G(V,E)$ or simply written as $G$. The graph labeling  was first introduced in 1960, it is a function that mapping integers or labels to graph elements (vertices, edges, or both of them) which must satisfy some certain criteria. This concepts was applied in some real problems. In 2007,  there was a new concept on labeling, i.e., ``vertex irregular total labeling" and ``edge irregular total labeling". In 2017, there was a new concept, i.e., ``vertex irregular reflexive labeling" and ``edge irregular reflexive labeling". The ``edge irregular reflexive $k-$ labeling" of $G$ is  a mapping that puts an even number label from $0$ to $2k_v$ to every vertex and a positive integer label from $1$ to $k_e$ to every edge with different weights for each edge. The minimum $k$ of the biggest label among all possible ``edge irregular reflexive $k-$ labeling" of $G$ is called the ``reflexive edge strength" of $G$, denoted as $res(G)$. There are only few result on ``vertex irregular reflexive labeling" or ``edge irregular reflexive labeling" of corona product  graphs. Hence, the purpose of this research are to investigate   $res$ of corona product of some ladder graphs and null graphs. Therefore, the goal of this study is to examine the $res$ of corona product of some ladder graphs and null graphs. We get the results  as follows: $res(SL_n \odot N_m)$ for ``$n \ge 2$ and $m \ge 1$"  are ``$\left \lceil \frac{2nm+3n-3}{3} \right \rceil$" for ``$2nm+3n-3 \not \equiv 2,3 \pmod{6}$" and ``$\left \lceil \frac{2nm+3n-3}{3} \right \rceil+1$" for ``$2nm+3n-3 \equiv 2,3 \pmod{6}$". Moreover, $res(DL_n \odot N_m)$ for $n\geq2$ and $m\geq1$ are ``$\left \lceil \frac{2nm+5n-4}{3} \right \rceil$" for ``$2nm+5n-4 \not \equiv 2,3 \pmod{6}$" and ``$\left \lceil \frac{2nm+5n-4}{3} \right \rceil+1$" for ``$2nm+5n-4 \equiv 2,3 \pmod{6}$". These results   contribute to developing the theory of reflexive labeling.