Abstract
A total labeling Φ : V (G) ∪ E(G) → {1, 2, ... , k} is called a vertex irregular total k-labeling of a graph G if different vertices in G have different weights. The weight of a vertex is defined as the sum of the labels of its incident edges and the label of that vertex. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper we deal with the total vertex irregularity strength of uniform theta graphs and centralized uniform theta graphs. Theta graph is a closer representation of bipolar electric or magnetic fields so labeling of various theta graphs can help the law of physics in future.
Highlights
A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s)
Such labelings were called irregular assignments and the irregularity strength s(G) of a graph G is known as the minimum k for which G has an irregular assignment using labels at most k
The main aim of this paper is to find an exact value of the total vertex irregularity strength of uniform theta graphs and centralized uniform theta graphs
Summary
A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). Such labelings were called irregular assignments and the irregularity strength s(G) of a graph G is known as the minimum k for which G has an irregular assignment using labels at most k This parameter as elaborated in [5], [6], [9], [11] and a detailed survey [12], has sought considerable attention of many a prolific author. Motivated by these papers, Bača et al in [8] introduced an edge irregular total k-labeling and a vertex irregular total k-labeling. The main aim of this paper is to find an exact value of the total vertex irregularity strength of uniform theta graphs and centralized uniform theta graphs
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