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

Let G = G(V, E) be a finite simple undirected graph with vertex set V and edge set E, where |E| and |V | 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, . . . , |V |} 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+(|E|− 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, . . . , |V | + |E|} with the property that for every edge xy ∈ E, the edge-weight set is equal to {f(x)+f(y)+ ∗ Corresponding author. † Also at Dept. Mathematics, University of West Bohemia, Pilsen, Czech Republic and Department of Informatics, King’s College London, U.K. 258 RAHMAWATI, SUGENG, SILABAN, MILLER AND BAČA f(xy) : x, y ∈ V, xy ∈ E} = {a, a+d, a+2d, . . . , a+(|E|−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, . . . , |V |}. 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 a |V | × |V | 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.