| Summary: | A total labeling <inline-formula> <tex-math notation="LaTeX">$\phi: V(G)\cup E(G) \to \{1,2, {\dots }, k\}$ </tex-math></inline-formula> is called a <italic>vertex irregular total</italic> <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula><italic>-labeling</italic> of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> if different vertices in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> for which the graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> has a vertex irregular total <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-labeling is called the <italic>total vertex irregularity strength</italic> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$tvs(G)$ </tex-math></inline-formula>. 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.
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