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Construction of new larger (a, d)-edge antimagic vertex graphs by using adjacency matrices

Abstract:

Let G = G(V,E) be a finite simple undirected graph with vertex set V and edge set E, where {pipe}E{pipe} and {pipe}V{pipe} are the number of edges and vertices on G. An (a, d)-edge antimagic vertex ((a, d)-EAV) labeling is a one-toone mapping f from V (G) onto {1, 2...,{pipe}{pipe}} with the property that for every edge xy ∈ E, the edge-weight set is equal to {f(x) + f(y): x, y ∈ V } = {a, a+d, a+2d, ..., a+({pipe}E{pipe}-1)d}, for some integers a > 0, d ≥ 0. An (a, d)-edge antimagic total ((a, d)-EAT) labeling is a one-toone mapping f from V ∪ E onto {1, 2,...,verbarVverbar +{pipe}E{pipe}} with the property that for every edge xy ∈ E, the edge-weight set is equal to {f(x)+f(y)+ f(xy): x, y ∈ V, xy ∈ E} = {a, a+d, a+2d,..., a+({pipe}E{pipe}-1)d}, where a > 0, d ≥ 0 are two fixed integers. Such a labeling is called a super (a, d)- edge antimagic total ((a, d)-SEAT) labeling if f(V) = {1, 2,...,{pipe}V{pipe}}. A graph that has an (a, d)-EAV ((a, d)-EAT or (a, d)-SEAT) labeling is called an (a, d)-EAV ((a, d)-EAT or (a, d)-SEAT) graph. For an (a, d)- EAV (or (a, d)-SEAT) graph G, an adjacency matrix of G is averbarVverbar ×verbarVverbar matrix AG = [aij ] such that the entry aij is 1 if there is an edge from vertex with index i to vertex with index j, and entry aij is 0 otherwise. This paper shows the construction of new larger (a, d)-EAV graph from an existing (a, d)-EAV graph using the adjacency matrix, for d = 1, 2. The results will be extended for (a, d)-SEAT graphs with d = 0, 1, 2, 3.