https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:37764 0⊆S₁⊆S₂⊆⋯ be defined as follows: S₁ is obtained from S₀ by adding all out-neighbors of vertices in S₀. For k⩾2, S_k is obtained from S_k−1 by adding all vertices w such that for some vertex v∈S_k−1, w is the unique out-neighbor of v in V∖S_k−1. We set M(S)=S₀∪S₁∪⋯, and call S a power dominating set for G if M(S)=V(G). The minimum cardinality of such a set is called the power domination number of G. In this paper, we determine the power domination numbers of de Bruijn and Kautz digraphs.]]> Wed 14 Apr 2021 12:21:11 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31448 Wed 11 Apr 2018 16:43:14 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:33940 Wed 04 Sep 2019 10:04:28 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:35653 Wed 02 Oct 2019 10:01:58 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:34786 zero forcing set if by iterating this process, all of the vertices in G become black. The zero forcing number of G is the minimum cardinality of a zero forcing set in G, and is denoted by Z(G). Davila and Kenter have conjectured in 2015 that Z(G)≥(g-3)(δ-2)+ δ where g and δ denote the girth and the minimum degree of G, respectively. This conjecture has been proven for graphs with girth g≤10. In this note, we present a proof for g≥5, d≥2, thereby settling the conjecture.]]> Tue 01 Sep 2020 11:52:36 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:37218 Thu 15 Apr 2021 12:01:59 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31515 v of a connected graph G (V, E) and a subset S of V, the distance between v and S is defined by d(v,S)=min{d(v,x):x∈S}. For an ordered k.-partition Π={S₁,S₂,…,S_k} of V, the representation of v with respect to Π is the k-vector r(v∣Π)=(d(v,S₁),d(v,S₂),…,d(v,S_k)). The k-partition Π is a resolving partition if the k-vectors r(v∣Π), v∈V are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. In this paper, we obtain the partition dimension of circulant graphs [formula cannot be replicated]]]> Sat 24 Mar 2018 08:43:35 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:17583 Sat 24 Mar 2018 08:03:58 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19436 Sat 24 Mar 2018 07:51:58 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:28278 metric basis. Metric dimension is the cardinality of a metric basis. A pair of vertices u, v is said to be strongly resolved by a vertex s, if there exists at least one shortest path from s to u passing through v, or a shortest path from s to v passing through u. A set W ⊆ V, is said to be a strong resolving set if for all pairs u, v ∉ W, there exists some element s ∈ W such that s strongly resolves the pair u, v. A strong resolving set of minimum cardinality is called a strong metric basis. The cardinality of a strong metric basis for G is called the strong metric dimension of G. The strong metric dimension (metric dimension) problem is to find a strong metric basis (metric basis) in the graph. In this paper, we solve the strong metric dimension and the metric dimension problems for the graph of tetrahedral diamond lattice.]]> Sat 24 Mar 2018 07:41:22 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:27706 spectrum. The energy of a graph is the sum of the absolute values of its eigenvalues. In this paper, we devise an algorithm which generates the adjacency matrix of WK - recursive structures WK(3,L) and WK(4,L) and use it in the effective computation of spectrum and energy.]]> Sat 24 Mar 2018 07:40:10 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:27616 Sat 24 Mar 2018 07:39:40 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:29436 Sat 24 Mar 2018 07:39:19 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:27629 γp(G). In this paper, we establish a fundamental result that would provide a lower bound for the power domination number of a graph. Further, we solve the power domination problem in polyphenylene dendrimers, Rhenium Trioxide (ReO3) lattices and silicate networks.]]> Sat 24 Mar 2018 07:34:26 AEDT ]]>