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...

A total edge-irregular k-labelling ξ : 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 edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-lab...

The total irregularity of a graph G is defined as 〖irr〗_t (G)=1/2 ∑_(u,v∈V(G))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈V(G). In this paper by using the Gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.

Martin Bača et al. [2] introduced the problem of determining the total vertex irregularity strengths of graphs. In this paper we discuss how the addition of new edge affect the total vertex irregularity strength.