Ali Ahmad, Martin Baca and Yasir Bashir: Total vertex irregularity strength of certain classes of unicyclic graphs, p.147-152

Abstract:

A total vertex irregular $k$-labeling $\phi$ of a graph $G$ is a labeling of the vertices and edges of $G$ with labels from the set $\{1,2, \dots, k\}$ in such a way that for any two different vertices $x$ and $y$ their weights $wt(x)$ and $wt(y)$ are distinct. Here, the weight of a vertex $x$ in $G$ is the sum of the label of $x$ and the labels of all edges incident with the vertex $x$. The minimum $k$ for which the graph $G$ has a vertex irregular total $k$-labeling is called the total vertex irregularity strength of $G$.

We have determined an exact value of the total vertex irregularity strength of certain classes of unicyclic graphs.

Key Words: Vertex irregular total $k$-labeling, total vertex irregularity strength, stars, kite graphs, paths, cycles.

2000 Mathematics Subject Classification: Primary: 05C78;
Secondary: 05C38.

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