Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Computing the Total $H$-irregularity Strength of Edge Comb Product of Graphs

Version 1 : Received: 12 August 2022 / Approved: 16 August 2022 / Online: 16 August 2022 (03:51:18 CEST)

How to cite: Wahyujati, M.F.; Susanti, Y. Computing the Total $H$-irregularity Strength of Edge Comb Product of Graphs. Preprints 2022, 2022080274 (doi: 10.20944/preprints202208.0274.v1). Wahyujati, M.F.; Susanti, Y. Computing the Total $H$-irregularity Strength of Edge Comb Product of Graphs. Preprints 2022, 2022080274 (doi: 10.20944/preprints202208.0274.v1).

## Abstract

A simple undirected graph $\Gamma=(V_\Gamma,E_\Gamma)$ admits an $H$-covering if every edge in $E_\Gamma$ belongs to at least one subgraph of $\Gamma$ that isomorphic to a graph $H$. For any graph $\Gamma$ admitting $H$-covering, a total Labelling $\beta: V_\Gamma \cup E_\Gamma \longrightarrow \{1,2,\dots,p\}$ is called an $H$-irregular total $p$-labelling of $\Gamma$ if every two different subgraphs $H_1$ and $H_2$ of $\Gamma$ isomorphic to $H$ have distinct weights where the weight $w_\beta(K)$ of subgraph $K$ of $\Gamma$ is defined as $w_f(K):=\displaystyle \sum_{v\in V_K} f(v) + \sum_{e\in E_K} f(e)$. The smallest number $p$ for which a graph $\Gamma$ admits an $H$-irregular total $p$-labelling is called the total $H$-irregularity strength of $\Gamma$ and is denoted by $ths(\Gamma)$. In this paper, we determine the total $H$-irregularity strength of edge comb product of two graphs.

## Keywords

Total H-Irregular Strength; Edge Comb Product Graph; Cycle; Path

## Subject

MATHEMATICS & COMPUTER SCIENCE, General Mathematics

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