Total Vertex-Irregularity Labelings for Subdivision of Several Classes of Trees

AbstractMotivated by the notion of the irregularity strength of a graph introduced by Chartrand et al. [3] in 1988 and various kind of other total labelings, Baca et al. [1] introduced the total vertex irregularity strength of a graph.In 2010, Nurdin, Baskoro, Salman and Gaos[5] determined the total vertex irregularity strength for various types of trees, namely complete k–ary trees, a subdivision of stars, and subdivision of particular types of caterpillars. In other paper[6], they conjectured that the total vertex irregularity strength of any tree T is only determined by the number of vertices of degree 1, 2, and 3 in T . In this paper, we attempt to verify this conjecture by considering a subdivision of several types of trees, namely caterpillars, firecrackers, and amalgamation of stars

Computing the Edge Irregularity Strengths of Chain Graphs and the Join of Two Graphs

In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V,E) with the vertex set V and the edge set E, a vertex k-labeling $\phi: V \rightarrow \{1,2,\dots, k\}$ is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their $w_\phi(e) \ne w_\phi(f)$, where the weight of an edge $e=xy \in E(G)$ is $w_\phi(xy)=\phi(x)+\phi(y)$. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research

The total irregularity strength of m copies of the friendship graph

This paper deals with the totally irregular total labeling of the disjoin union of friendship graphs. The results shows that the disjoin union of  copies of the friendship graph is a totally irregular total graph with the exact values of the total irregularity strength equals to its edge irregular total strength