Total Edge Irregularity Strength for Graphs

TL;DR

This paper investigates the total edge irregularity strength of graphs, providing exact values for both connected and disconnected cases, advancing understanding of graph labelings with unique edge weights.

Contribution

It determines the total edge irregularity strength for both connected and disconnected graphs, filling a gap in graph labeling theory.

Findings

Exact total edge irregularity strength for connected graphs

Exact total edge irregularity strength for disconnected graphs

New bounds and formulas for graph labelings

Abstract

An edge irregular total $k$-labelling $f : V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ of a graph $G$ is a labelling of the vertices and the edges of $G$ in such a way that any two different edges have distinct weights. The weight of an edge $e$, denoted by $wt(e)$, is defined as the sum of the label of $e$ and the labels of two vertices which incident with $e$, i.e. if $e=vw$, then $wt(e)=f(e)+f(v)+f(w)$. The minimum $k$ for which $G$ has an edge irregular total $k$-labelling is called the total edge irregularity strength of $G.$ In this paper, we determine total edge irregularity of connected and disconnected graphs.