https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 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:28963 Thu 26 Jul 2018 16:56:33 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:29729 G = (V,E) and for a pair of vertices u, v ∈ V , we say that a vertex w ∈ V resolves u and v if the shortest path from w to u is of a different length than the shortest path from w to v. A set of vertices R ⊆ V is a resolving set if for every pair of vertices u and v in G, there exists a vertex w ∈ R that resolves u and v. The minimum weight resolving set problem is to find a resolving set M for a weighted graph G such that Σ_{v∈ M} w(v) is minimum, where w(v) is the weight of vertex v. In this paper, we explore the possible solutions of this problem for grid graphs P_n⎕P_m where 3 ≤ n ≤ m. We give a complete characterization of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is 2n − 2. We show that the grid has the property that given a landmark set, we only need to investigate whether or not all pairs of vertices that share common neighbors are resolved to determine if the whole graph is resolved. We use this result to provide a characterization of a class of minimals whose cardinalities range from 4 to 2n−2 and show that the number of such minimals is Ω(2ⁿ).]]> Sat 24 Mar 2018 07:37:31 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 ]]>