Total Edge Irregularity Strength for Graphs

Abstract

An edge irregular total $k$-labelling $f:V(G)\cup E(G)\to \{1,2,\dots ,k\}$ 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.

there is a gap in the proof, needs to be addressed properly