Abstract
A vertex (edge) irregular total k-labeling ? of a graph G is a labeling of the vertices and edges of G with labels from the set {1,2,...,k} in such a way that any two different vertices (edges) have distinct weights. 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, whereas the weight of an edge is the sum of label of the edge and the vertices incident to that edge. The minimum k for which the graph G has a vertex (edge) irregular total k-labeling is called the total vertex (edge) irregularity strength of G. In this paper, we are dealing with infinite classes of convex polytopes generated by prism graph and antiprism graph. We have determined the exact value of their total vertex irregularity strength and total edge irregularity strength.
Highlights
The graph labeling has caught the attention of many authors and many new labeling results appear every year
In the two theorems, we find the total vertex irregularity strength and total edge irregularity strength of the graph of convex polytope Sn
In the two theorems we find the total vertex irregularity strength and total edge irregularity strength of the graph of convex polytope Tsn
Summary
The graph labeling has caught the attention of many authors and many new labeling results appear every year. In [2] Baca et al defined a new graph invariant, called the total vertex (edge) irregularity strength of G, tvs(G) (tes(G)), that is the minimum k for which the graph G has a vertex (edge) irregular total k-labeling. Nierhoff [16] proved that for every (p, q)-graph G (i.e. the graph with p vertices and q edges) with no component of order at most 2 and G 6= K3, the irregularity strength s(G) ≤ p − 1 From this result it follows that (1.1). We determined an exact value of total vertex irregularity strength as well as total edge irregularity strength for some infinite classes of convex polytopes
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