Modular Irregularity Strength of Triangular Book Graph

TL;DR

This paper investigates the modular irregularity strength of triangular book graphs, establishing that they admit a modular irregular labeling with equal irregularity and modular irregularity strengths, except for specific small cases.

Contribution

It introduces the concept of modular irregularity strength for graphs and determines this value for triangular book graphs, extending the understanding of graph labelings.

Findings

Triangular book graphs admit a modular irregular labeling.

The modular irregularity strength equals the irregularity strength for these graphs.

Except for a small case, the two strengths are equal.

Abstract

This paper deals with the modular irregularity strength of a graph of n vertices, a new graph invariant, modified from the irregularity strength, by changing the condition of the vertex-weight set associate to the well-known irregular labeling from n distinct positive integer to Z_n-the group of integer modulo n. Investigating the triangular book graph B_m^((3)), we first find the irregularity strength of triangular book graph s(B_m^((3)) ), as the lower bound for the modular irregularity strength, and then construct a modular irregular s(B_m^((3)) )-labeling. The result shows that triangular book graphs admit a modular irregular labeling and its modular irregularity strength and irregularity strength are equal, except for a small case and the infinity property.