https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:44976 G, an edge labeling f_e : E(G) → {1, 2, . . . , k_e} and a vertex labeling f_v : V(G) → {0, 2, 4, . . . , 2k_v} are called total k-labeling, where k = max{k_e, 2k_v}. The total k-labeling is called an edge irregular reflexive k-labeling of the graph G, if for every two different edges xy and x′ y′ of G, one has wt(xy) = f_v(x) + f_e(xy) + f_v(y) ̸= wt(x′ y′) = f_v(x′) + f_e(x′ y′) + f_v(y′). The minimum k for which the graph G has an edge irregular reflexive k-labeling is called the reflexive edge strength of G. In this paper we determine the exact value of the reflexive edge strength for cycles, Cartesian product of two cycles and for join graphs of the path and cycle with 2K₂.]]> Wed 26 Oct 2022 08:53:34 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:43506 Wed 21 Sep 2022 10:29:20 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:40960 Wed 20 Jul 2022 16:53:18 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:30480 G = (V,E) admits an H-covering if every edge in E is contained in a subgraph H’= (V’, E’) of G which is isomorphic to H. In this case we say that G is H-supermagic if there is a bijection f : V ⋃ E → {1,...,|V| + |E|} such that f(V) = {1,...,|V|} and ∑_vϵV(H')f(v)+∑_vϵV(H')f(e) is constant over all subgraphs H' of G which are isomorphic to H. Extending results from [M. Roswitha and E.T. Baskoro, Amer. Inst. Physics Conf. Proc. 1450 (2012), 135-138], we show that the firecracker F_k,n is F_2,n-supermagic, the banana tree B_k,n is B_k-1,n-supermagic and the flower F_n is C₃-supermagic.]]> Wed 11 Apr 2018 14:06:23 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:30428 G we define k-labeling ρ such that the edges of G are labeled with integers {1, 2, . . . , k_e} and the vertices of G are labeled with even integers {0, 2, . . . , 2k_v}, where k = max{k_e, 2k_v}. The labeling ρ is called an edge irregular k-labeling if distinct edges have distinct weights, where the edge weight is defined as the sum of the label of that edge and the labels of its ends. The smallest k for which such labeling exist is called the reflexive edge strength of G. In this paper we give exact values of reflexive edge strength for prisms, wheels, baskets and fans.]]> Wed 11 Apr 2018 13:07:11 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:4488 Wed 11 Apr 2018 12:54:05 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:14336 Wed 11 Apr 2018 12:35:12 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:26991 Wed 11 Apr 2018 11:44:28 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:30604 Wed 11 Apr 2018 10:30:31 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:34673 Wed 10 Apr 2019 16:58:07 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:33940 Wed 04 Sep 2019 10:04:28 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:35653 Wed 02 Oct 2019 10:01:58 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:52926 Wed 01 Nov 2023 09:36:11 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:38837 Tue 15 Feb 2022 14:02:29 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:32665 Tue 10 Jul 2018 10:51:11 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:36069 Tue 04 Feb 2020 11:12:06 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:22237 G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V(G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n²) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.]]> Tue 04 Feb 2020 10:56:52 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:23057 Tue 04 Feb 2020 10:56:25 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:31129 Tue 04 Feb 2020 10:56:06 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:43783 G we define k-labeling ρ such that the edges of G are labeled with integers {1,2,…,k_e} and the vertices of G are labeled with even integers {0,2,…,2k_v}, where k=max{k_e,2k_v}. The labeling ρ is called a vertex irregular reflexive k-labeling if distinct vertices have distinct weights, where the vertex weight is defined as the sum of the label of that vertex and the labels of all edges incident this vertex. The smallest k for which such labeling exists is called the reflexive vertex strength of G.]]> Thu 29 Sep 2022 13:48:03 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:51055 Thu 17 Aug 2023 10:22:44 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:43331 Thu 15 Sep 2022 14:57:41 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:7847 Sat 24 Mar 2018 10:48:16 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:31775 Sat 24 Mar 2018 08:44:16 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:7763 Sat 24 Mar 2018 08:41:57 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:7822 Sat 24 Mar 2018 08:37:38 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:8136 Sat 24 Mar 2018 08:36:10 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:7645 t, and girth (length of shortest cycle) at least g ≥ t + 1. In 1975, Erdős proposed the problem of determining the extremal numbers ex(n;4) of a graph of n vertices and girth at least 5. In this paper, we consider a generalized version of this problem, for t ≥ 5. In particular, we prove that ex(n;6) for n = 29, 30 and 31 is equal to 45, 47 and 49, respectively.]]> Sat 24 Mar 2018 08:35:58 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:7252 Sat 24 Mar 2018 08:33:50 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:17169 Sat 24 Mar 2018 08:06:31 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:19509 Sat 24 Mar 2018 08:02:18 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:16933 Sat 24 Mar 2018 08:00:30 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:21608 G = (V,E) is a bijection from the set of edges to the set of integers {1, 2,..., ∣E∣}. The weight of a vertex v is the sum of the labels of all the edges incident with v. If the vertex weights are all distinct then we say that the labeling is vertex-antimagic, or simply, antimagic. A graph that admits an antimagic labeling is called an antimagic graph. In this paper, we present a new general method of constructing families of graphs with antimagic labelings. In particular, our method allows us to prove that generalized web graphs and generalized flower graphs are antimagic.]]> Sat 24 Mar 2018 07:59:32 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:20870 Sat 24 Mar 2018 07:57:58 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:19216 Sat 24 Mar 2018 07:54:58 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:16253 Sat 24 Mar 2018 07:54:13 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:21755 Sat 24 Mar 2018 07:53:08 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:19305 Sat 24 Mar 2018 07:49:59 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:28279 f(H') = Σ_v∈(H') f(v)+Σ_e∈(H') f(e) form an arithmetic progression with the initial term a and the common difference d. When f(V) = {1, 2,...,⏐V⏐}, then G is said to be super (a, d)-H-antimagic. In this paper, we study super (a, d)-H-antimagic labellings of a disjoint union of graphs for d = ⏐E(H)⏐ - ⏐V(H)⏐.]]> Sat 24 Mar 2018 07:41:22 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:28280 m,n is an example of join graphs and we give an antimagic labeling for K_m,n,n≥2m+1. In this paper we also provide constructions of antimagic labelings of some complete multipartite graphs.]]> Sat 24 Mar 2018 07:41:22 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:26506 Sat 24 Mar 2018 07:35:33 AEDT ]]> https://nova.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 ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:30337 Sat 24 Mar 2018 07:31:47 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:28307 Discrete Math. 307 (2007), 2925-2933] showed the cyclic-magic and cyclic-supermagic behaviour of several classes of connected graphs. They discussed cycle-magic labellings of subdivided wheels and friendship graphs, but there are no further results on cycle-magic labellings of other families of subdivided graphs. In this paper, we find cycle-magic labellings for subdivided graphs. We show that if a graph has a cycle-(super)magic labelling, then its uniform subdivided graph also has a cycle-(super)magic labelling. We also discuss some cycle-supermagic labellings for nonuniform subdivided fans and triangular ladders.]]> Sat 24 Mar 2018 07:27:06 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:23536 Sat 24 Mar 2018 07:16:59 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:22311 antimagic labeling of a graph, that is, a vertex antimagic edge labeling and they also conjectured that every connected graph, except K₂, is antimagic. As a means of providing an incremental advance towards proving the conjecture of Hartsfield and Ringel, in this paper we provide constructions whereby, given any degree sequence pertaining to a tree, we can construct two different vertex antimagic edge trees with the given degree sequence. Moreover, we modify a construction presented for trees to obtain an antimagic unicyclic graph with a given degree sequence pertaining to a unicyclic graph.]]> Sat 24 Mar 2018 07:14:43 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:32758 sum graph G is a graph with an injective mapping of the vertex set of G onto a set of positive integers S in such a way that two vertices of G are adjacent if and only if the sum of their labels is an element of S. In an exclusive sum graph the integers of S that are the sum of two other integers of S form a set of integers that label a collection of isolated vertices associated with the graph G. A graph bears a k-exclusive sum labelling (abbreviated k-ESL), if the set of isolated vertices is of cardinality k, an optimal exclusive sum labelling, if k is as small as possible, and Δ-optimal if k equals the maximum degree of the graph. In this paper, observing that the property of having a k-ESL is hereditary, we provide a characterisation of graphs that have a k-exclusive sum labelling, for any k ≥ 1, in terms of describing a universal graph for the property.]]> Mon 23 Jul 2018 11:04:20 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:47569 Mon 23 Jan 2023 13:39:29 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:49558 Mon 22 May 2023 09:13:08 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:42342 Mon 22 Aug 2022 13:54:35 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:47958 Mon 13 Feb 2023 14:27:59 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:44150 Mon 10 Oct 2022 09:24:19 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:45866 Mon 07 Nov 2022 15:55:14 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:42816 Mon 05 Sep 2022 11:14:40 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:34725 Fri 26 Apr 2019 15:59:25 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:46572 Fri 25 Nov 2022 11:47:29 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:53858 Fri 19 Jan 2024 12:30:50 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:23800 order/degree problem, that is, to determine the smallest diameter of a digraph given order and maximum out-degree. There is no general efficient algorithm known for the construction of such optimal digraphs but various construction techniques for digraphs with minimum diameter have been proposed. In this paper, we survey the known techniques.]]> Fri 06 Oct 2023 15:41:42 AEDT ]]>