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Abstract. Let = ( , ) be a graph. A total labeling : ∪ → {1, 2, ⋯ , } is called a totally irregular total -labeling of if every two distinct vertices and in satisfy ( ) ≠ ( ) and every two distinct edges 1 2 and 1 2 in satisfyThe minimum for which a graph has a totally irregular total -labeling is called the total irregularity strength of , denoted by ( ). In this paper, we consider an upper bound on the total irregularity strength of copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total -labeling of a regular graph and we consider the total irregularity strength of copies of a path on two vertices, copies of a cycle, and copies of a prism 2 .

Let G = (V (G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V (G) ∪ E(G) → {1,2...,k}. The vertex weight v under the labeling f is denoted by Wf(v) and defined by Wf(v) = f(v) + Σuv∈E(G)f(uv). A total k-labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k such that G has a vertex irregular total k-labeling. This labeling was introduced by Bača, Jendrol', Miller, and Ryan in 2007. Let G and H be two connected graphs. Let o be a vertex of H . The comb product between G and H, in the vertex o, denoted by G⊳o H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. In this paper, we determine the total vertex irregularity strength of comb product of Cn and Cm where m ∈ {1,2}.

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a graph and <em>k</em> be a positive integer. A total <em>k</em>-labeling of <em>G</em> is a map <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,<em>k</em>}. The vertex weight <em>v</em> under the labeling <em>f</em> is denoted by w_<em>f</em>(<em>v</em>) and defined by <em>w</em>_<em>f</em>(<em>v</em>) = <em>f</em>(<em>v</em>) + \sum_{uv \in{E(G)}} {<em>f</em>(<em>uv</em>)}. A total <em>k</em>-labeling of <em>G</em> is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of <em>G</em>, denoted by <em>tvs</em>(<em>G</em>), is the minimum <em>k</em> such that <em>G</em> has a vertex irregular total <em>k</em>-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let <em>G</em> and <em>H</em> be two connected graphs. Let <em>o</em> be a vertex of <em>H</em>. The comb product between <em>G</em> and <em>H</em>, denoted by <em>G </em>\rhd_o <em>H</em>, is a graph obtained by taking one copy of <em>G</em> and |<em>V</em>(<em>G</em>)| copies of <em>H</em> and grafting the i-th copy of <em>H</em> at the vertex <em>o</em> to the i-th vertex of <em>G</em>. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

Let G = (V(G), E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f: V(G) ∪ E(G) → {1,2, ⋯, k }. The edge weight uv under the labeling f is defined by w f (uv) = f(u) + f(uv) + f(v) and denoted by by w f (uv) and. A total k-labeling of G is called edge irregular if every two distinct edges have distinct weight. The total edge irregularity strength of G is denoted by tes(G) and defined by the minimum k such that G has an edge irregular total k-labeling. The labeling was introduced by Bača et al. in 2007. In this paper, we determine the total edge irregularity strength of some copies of ladder graphs.