https://nova.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:30377 α . We prove that all the eigenvalues of ⌈ (including multiplicities) are, in addition, eigenvalues of ⌈^α. This study is carried out by introducing several reduced matrix representations of ⌈^α. As an example of our techniques, we study some basic properties of the Alegre digraph and its base.]]> Wed 11 Apr 2018 14:19:34 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:22239 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)+ 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 ∣V∣ × ∣V∣ matrix A_G = [a_ij] such that the entry a_ij 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.]]> Wed 11 Apr 2018 13:24:44 AEST ]]>