# TOTAL VERTEX IRREGULARITY STRENGTH OF LADDER RELATED GRAPHS.

Byline: Ashfaq Ahmad, Syed Ahtsham ul Haq Bokhary, Roslan Hasni and Slamin

1. INTRODUCTION AND DEFINITIONSAs a standard notation, assume that G = G (V, E) is a finite, simple and undirected graph with p vertices and q edges. A labeling of a graph is any mapping that sends some set of graph elements to a set of numbers (usually positive integers). If the domain is the vertex-set or the edge-set, the labeling are called respectively vertex-labeling or edge-labeling. If the domain is V E then we call the labeling atotal labeling. In many cases it is interesting to consider thesum of all labels associated with a graph element. This will be called the weight of element.Motivated by total labeling mentioned in a book of Wallis [10], Baca et al. in [5] introduced a vertex irregular total labeling of graphs. For a simple graph G = (V, E) withvertex set V and edge set E, a labeling : V E {1, 2, . . ., k} is called total k-labeling. The associated vertex weightof a vertex x V (G) under a total k-labeling is defined aswhere N(x) is the set of neighbors of x. A total k-labeling is defined to be a vertex irregular total labeling of a graph Gif for every two different vertices x and y of G,wt(x) wt(y)The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G).In this paper, we study properties of the vertex irregular total labeling and determine a value of the total vertex irregularity strength for classes of ladder related graphs, such as triangular ladder, diagonal ladder, triangular snake anddouble triangular snake.Triangular Ladder, denoted by TLn, is the graph obtained from ladder by adding single diagonal to each rectangle.Thus the vertex set of TLn is {vi,j |1 = i = 2, 1 = j = n} and the edge set of Ln isj = n} {vi,jvi+1,j+1|i = 1, 1 = j = n - 1}Diagonal Ladder, denoted by DLn, is the graph obtainedfrom ladder by adding two diagonals to each rectangle. Thusthe vertex set of DLn is {vi,j |1 = i = 2, 1 = j = n} and the edge set of DLn isE (G) = {vi,jvi,j+1|1 = i = 2, 1 = j = n-1}{vi,jvi+1,j |i = 1, 1 = j= n}{vi,jvi+1,j+1|i = 1, 1 = j = n - 1} {vi,jvi+1,j-1|i = 1, 2 = j = n}Triangular Snake, denoted by TSn, is the graph obtainedfrom a non-trivial path Pn: v1, v2, . . . , vn by adding newvertices u1, u2, . . . , un-1 joining each ui with vi and vi+1 (1 = i= n - 1). Thus the vertex set of TSn is

{vi, uj|1 = i = n, 1 = j = n - 1}

and the edge set of TSn is

{vivi+1, viui, uivi+1|1 = i = n - 1}

Double Triangular Snake, denoted by DTSn, is the graph obtained from a triangular snake TSn by adding new vertices w1,w2, . . . ,wn-1 joining each wi with vi and vi+1 (1 = i = n - 1). Thus the vertex set of DTSn is{vi, uj,wj |1 = i = n, 1 = j = n - 1}and the edge set of DTSn is{vivi+1, viui, uivi+1, viwi,wivi+1|1 = i = n - 1}

2. KNOWN RESULTS

The following theorem proved in [5], establishes lower and upper bound for the total vertex irregularity strength of a (p, q)-graph.

Theorem 1 [5] Let G be a (p, q)-graph with minimum degree d = d (G) and maximum degree = (G). ThenFor a regular Hamiltonian (p, q) graph G, it was showed in[5] that tvs(G) p 2 . Thus for cycle Cp we have thatRecently, a much stronger upper bound on total vertexirregularity strength of graphs has been established in [4]. In6, 7, 8], Nurdin et al. found the exact values of total vertex irregularity strength of trees, several types of trees and disjoint union of t copies of path. Whereas the total vertexirregularity strength of cubic graphs, wheel related graphs,Jahangir graph Jn,2 for n = 4 and circulant graph Cn(1, 2) for n = 5 has been determined by Ahmad et al. [1, 2, 3]. K. Wijaya et al. [11, 12] found the exact value of the totalvertex irregularity strength of wheels, fans, suns, friendship, and complete bipartite graphs. Slamin et al. [9] determined the total vertex irregularity strength of disjoint union of sungraphs.3. MAIN RESULTWe start this section with the result on the total vertexirregularity strength of triangular ladder TLn graph in the following theorem.