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 vertical bar E vertical bar and vertical bar V vertical bar are the number of edges and vertices on G. An (a,d)-edge antimagic vertex ((a,d)-EAV) labeling is a one-to-one mapping f from V (G) onto {1, 2,..., vertical bar V vertical bar} with the property that for every edge xy is an element of E, the edge-weight set is equal to {f(x) + f(y) : x, y is an element of V} = {a, a+d, a+2d,..., a+(vertical bar E vertical bar- 1) d}, for some integers a > 0, d >= 0. An (a, d)-edge antimagic total ((a,d)-EAT) labeling is a one-to-one mapping f from V boolean OR E onto {1, 2,..., vertical bar V vertical bar + vertical bar E vertical bar} with the property that for every edge xy is an element of E, the edge-weight set is equal to {f(x)+ f(y)+f(xy) : x, y is an element of V, xy is an element of E} = {a, a+d, a+2d,..., a+(vertical bar E vertical bar- 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,..., vertical bar V vertical bar}. 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 vertical bar V vertical bar x vertical bar V vertical bar matrix AG = [a(ij)] such that the entry aij is 1 if there is an edge from vertex with index i to vertex with index j, and entry a(ij) 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.