Some Types of Irregular Labeling of Diamond Networks on Ten Vertices

There are three interesting parameters in irregular networks based on total labelling, i.e. the total vertex irregularity strength, the total edge irregularity strength, and the total irregularity strength of a graph. Besides that, there is a parameter based on edge labelling, i.e., the irregular labelling. In this paper, we determined the four parameters for diamond graph on eight vertices.

Total edge irregularity strength of some cycle related graphs

<p>An edge irregular total <em>k</em>-labeling <em>f</em> : <em>V</em> ∪ <em>E</em> → 1,2, ..., <em>k</em> of a graph <em>G</em> = (<em>V,E</em>) is a labeling of vertices and edges of <em>G</em> in such a way that for any two different edges <em>uv</em> and <em>u'v'</em>, their weights <em>f</em>(<em>u</em>)+<em>f</em>(<em>uv</em>)+<em>f</em>(<em>v</em>) and <em>f</em>(<em>u'</em>)+<em>f</em>(<em>u'v'</em>)+<em>f</em>(<em>v'</em>) are distinct. The total edge irregularity strength tes(<em>G</em>) is defined as the minimum <em>k</em> for which the graph <em>G</em> has an edge irregular total <em>k</em>-labeling. In this paper, we determine the total edge irregularity strength of new classes of graphs <em>C_m</em> @ <em>C_n</em>, <em>P_m,n</em>* and <em>C_m,n</em>* and hence we extend the validity of the conjecture tes(<em>G</em>) = max {⌈|<em>E</em>(<em>G</em>)|+2)/3⌉, ⌈(Δ(<em>G</em>)+1)/2⌉}<em> </em> for some more graphs.</p>

On the total irregularity strength of convex polytope graphs

A vertex (edge) irregular total k-labeling ? of a graph G is a labeling of the vertices and edges of G with labels from the set {1,2,...,k} in such a way that any two different vertices (edges) have distinct weights. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x, whereas the weight of an edge is the sum of label of the edge and the vertices incident to that edge. The minimum k for which the graph G has a vertex (edge) irregular total k-labeling is called the total vertex (edge) irregularity strength of G. In this paper, we are dealing with infinite classes of convex polytopes generated by prism graph and antiprism graph. We have determined the exact value of their total vertex irregularity strength and total edge irregularity strength.

Computing Total Edge Irregularity Strength for Heptagonal Snake Graph and Related Graphs

A labeling of edges and vertices of a simple graph 𝐺(𝑉,𝐸) by a mapping Ŧ:𝑉(𝐺) ∪ 𝐸(𝐺)→{ 1,2,3,…,Ћ} provided that any two pair of edges have distinct weights is called an edge irregular total Ћ-labeling. If Ћ is minimum and 𝐺 admits an edge irregular total Ћ -labelling, then Ћ is called the total edge irregularity strength (TEIS) and denoted by 𝑡𝑒𝑠(𝐺). In this paper, the definitions of the heptagonal snake graph HPSn ,the double heptagonal snake graph 𝐷(HPSn) and an 𝑙−multiple heptagonal snake graph 𝐿(HPSn) have been introduced. The exact value of TEISs for the new family has also been investigated.

Total Edge Irregularity Strength of q Tuple Book Graphs

Let G(V, E) be a simple, undirected, and finite graph with a vertex set V and an edge set E. An edge irregular total k-labelling is a function f from the set V \cup E to the set of non-negative integer set (1, 2, ... , k) such that any two different edges in E have distinct weights. The weight of edge xy is defined as the sum of the label of vertex x, the label of vertex y and the label of edge xy. The minimum k for which the graph G can be labelled by an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of triple book graphs, quadruplet book graphs and quintuplet book graphs. In this paper, we construct an edge irregular total of k-labelling and show the exact value of the total edge irregularity strength of q tuple book graphs.