Further study on edge trimagic total labeling

Abstract

Graph labeling is an injective mapping from elements of a graph, the newlinevertices, edges or both to a set of numbers (usually positive integers). If the newlinedomain of the mapping is the set of vertices (or edges) then the labeling is newlinecalled vertex labeling (or edge labeling), and if the domain of the mapping is newlinethe set of vertices and edges, then the label becomes total labeling. An edge newlinetrimagic total labeling of a graph G = (V, E) with p vertices and q edges is a newlinebijection f: V(G) E(G) and#8594; {1, 2, , p + q} such that for each edge uv newlineE(G), the value of f(u) + f (uv) + f(v) is either k1 or k2 or k3. An edge trimagic newlinetotal labeling is called as a super edge trimagic total labeling, if the vertices are newlinelabeled with the smallest possible integers i.e. 1, 2, , p. In this thesis, we newlineinvestigate the edge trimagic total labeling for Umbrella Un, m, Total graph newlineT(Pn), Mobius Ladder Mn, Book Bn, Grid Pm m Pn, Dumb bell Dbn (for all n and#8805; 3), newlineClosed helm CHn (for all n and#8805; 3), Antiprism An (for all n and#8805; 3), Square graph of newlineCn (even n and#8805; 4), Circular Ladder graph CL(n) (for all n and#8805; 3), (n, l) - dragon and newlineMusical graph M2n (n and#8805; 3). The above mentioned graphs also admit super edge newlinetrimagic total labeling. The shadow and splitting graphs of star, path and cycle newlineare also proved to admit an edge trimagic total labeling and super edge trimage newlinetotal labeling. An edge trimagic total labeling is called a superior edge trimagic newlinetotal labeling, if the edges are labeled with the smallest possible integers i.e. 1, newline2, , q. The thesis explores edge trimagic total labeling and superior edge newlinetrimagic total labeling for the graphs such as Coconut tree T(n, m), Windmill graph Wn(m) and Jelly fish graph J(m, n). We have also examined edge trimagic newlinetotal labeling of vertex duplication of graphs. We proved that duplication of an newlinearbitrary vertex in cycle Cn, duplication of a vertex of degree 3 and central newlinevertex of wheel Wn and duplication of central vertex of fan Fn admit edge newlinetrimagic total labeling and super edge trimagic total l