A vertex-irregular total k-labelling λ : V (G)∪E(G) −→ {1, 2, ..., k} of a graph G is a labelling of vertices and edges of G in such a way that for any different vertices x and y, their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x. The minimum k for which a graph G has a vertex-irregular total k-lab...

Consider a simple graph $G$. We call labeling $w:E(G)\cup V(G)\rightarrow \{1, 2, \dots, s\}$ (\textit{total vertex}) \textit{product-irregular}, if all product degrees $pd_G(v)$ induced by this are distinct, where $pd_G(v)=w(v)\times\prod_{e\ni v}w(e)$. The strength of $w$ is $s$, the maximum number used to label members $E(G)\cup V(G)$. minimum value $s$ that allows some irregular called \tex...

In this note a new measure of irregularity of a simple undirected graph G is introduced. It is named the total irregularity of a graph and is defined as irrt(G) = 1 2 ∑ u,v∈V (G) |dG(u)− dG(v)| , where dG(u) denotes the degree of a vertex u ∈ V (G). The graphs with maximal total irregularity are determined. It is also shown that among all trees of same order the star graph has the maximal total...

Let G = (V;E) be a graph. A total labeling ψ : V ⋃ E → {1, 2, ....k} is called totally irregular k-labeling of if every two distinct vertices u and v in (G) satisfy wt(u) ≠wt(v); edges u1u2 v1v2 E(G) wt(u1u2) ≠ wt(v1v2); where (u) + ∑uv∊E(G) ψ(uv) ψ(u1) ψ(u1u2) ψ(u2): The minimum k for which graph has the irregularity strength G, denoted by ts(G): In this paper, we determine exact value cubic g...

<abstract><p>We introduce the general Albertson irregularity index of a connected graph $ G and define it as A_{p}(G) = (\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}} $, where p is positive real number d(v) degree vertex v in $. The new not only generalization well-known \sigma $-index, but also Minkowski norm vertex. We present lower upper bounds on index. In addition, we study ext...

Albertson [3] has defined the irregularity of a simple undirected graph G = (V,E) as irr(G) = ∑ uv∈E |dG(u)− dG(v)| , where dG(u) denotes the degree of a vertex u ∈ V . Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [13]. For general graphs with n vertices...