On Total Vertex Irregularity Strength of Honeycomb Derived Networks

Let \( G(V, E) \) be a simple graph. For a labeling \( \partial: V \cup E \to \{1, 2, 3, \dots, k\} \), the weight of a vertex \( x \) is defined as

\[
wt(x) = \partial(x) + \sum_{xy \in E} \partial(xy).
\]

The labeling \( \partial \) is called a vertex irregular total \( k \)-labeling if for every pair of distinct vertices \( x \) and \( y \), \( wt(x) \neq wt(y) \). The minimum \( k \) for which the graph \( G \) has a vertex irregular total \( k \)-labeling is called the total vertex irregularity strength of \( G \) and is denoted by \( tvs(G) \). In this paper, we obtain a bound for the total vertex irregularity strength of honeycomb and honeycomb derived networks.