https://nova.newcastle.edu.au/vital/access/manager/Index ${session.getAttribute("locale")} 5 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:26288 v∈N(v)l(v) + d(u), where d(u) denotes the degree of u and N(u) denotes the open neighborhood of u. In this paper we introduce a new labeling called d-lucky labeling and study the same as a vertex coloring problem. We define a labeling l as d-lucky if c(u) ≠ c(v), for every pair of adjacent vertices u and v in G. The d-lucky number of a graph G, denoted by η_dl(G), is the least positive k such that G has a d-lucky labeling with {1,2,...,k} as the set of labels. We obtain η_dl(G) = 2 for hypercube network, butterfly network, benes network, mesh network, hypertree and X-tree.]]> Wed 11 Apr 2018 16:49:46 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4683 Wed 11 Apr 2018 16:38:25 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4682 Wed 11 Apr 2018 15:33:12 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:17117 Wed 11 Apr 2018 14:48:45 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:9198 0, that is, (Δ,D,−ε)-graphs. The parameter ε is called the defect. Graphs of defect 1 exist only for Δ = 2. When ε > 1, (Δ,D,−ε)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a (Δ,D,−2)-graph with Δ ≥ 4 and D ≥ 4 is 2D. Second, and most important, we prove the non-existence of (Δ,D,−2)-graphs with even Δ ≥ 4 and D ≥ 4; this outcome, together with a proof on the non-existence of (4, 3,−2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,−ε)-graphs with D ≥ 2 and 0 ≤ ε ≤ 2. Such a catalogue is only the second census of (Δ,D,−2)-graphs known at present, the first being the one of (3,D,−ε)-graphs with D ≥ 2 and 0 ≤ ε ≤ 2 [14]. Other results of this paper include necessary conditions for the existence of (Δ,D,−2)-graphs with odd Δ ≥ 5 and D ≥ 4, and the non-existence of (Δ,D,−2)-graphs with odd Δ ≥ 5 and D ≥ 5 such that Δ ≡ 0, 2 (mod D). Finally, we conjecture that there are no (Δ,D,−2)-graphs with Δ ≥ 4 and D ≥ 4, and comment on some implications of our results for the upper bounds of N(Δ,D).]]> Wed 11 Apr 2018 12:46:35 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:17695 Wed 11 Apr 2018 11:41:51 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4598 Wed 11 Apr 2018 11:37:35 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7829 Wed 11 Apr 2018 11:18:43 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:5502 Wed 11 Apr 2018 10:35:09 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4975 Wed 11 Apr 2018 09:35:56 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: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:512 = 41, cannot have more than 2d vertices. We show that 2d is the best possible upper bound by constructing planar digraphs of diameter 2 having exactly 2d vertices. Furthermore, we give upper and lower bounds for the largest possible order of planar digraphs with diameter greater than 2.]]> Thu 25 Jul 2013 09:10:26 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:379 Thu 25 Jul 2013 09:10:05 AEST ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9241 Sat 24 Mar 2018 11:12:48 AEDT ]]> 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: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://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9412 Sat 24 Mar 2018 08:39:32 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7792 Sat 24 Mar 2018 08:39:20 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7828 Sat 24 Mar 2018 08:37:39 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7826 Sat 24 Mar 2018 08:37:39 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7820 Sat 24 Mar 2018 08:37:36 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9144 Sat 24 Mar 2018 08:37:28 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:8137 Sat 24 Mar 2018 08:36:09 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:8153 Sat 24 Mar 2018 08:36:06 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:7781 0 and t ≥ 0 such that EDp+t(G) = EDt(G) [EDp+t(G) ≅ EDt(G)]. We derive that almost all digraphs have period two and tail zero. On the other hand, we show that there exist digraphs with arbitrarily large iso-period. We construct them using the generalized lexicographic product, taking as a basis an odd cycle graph. Additionally, we determine the period and tail of any tree, as well as other classes of graphs, by considering the connectivity of a certain subdigraph of ED(G) that 'concentrates' the eccentricities of all vertices of G.]]> Sat 24 Mar 2018 08:35:30 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:1521 Sat 24 Mar 2018 08:30:52 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:1518 Sat 24 Mar 2018 08:30:47 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:15630 Sat 24 Mar 2018 08:23:46 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:11177 Sat 24 Mar 2018 08:14:32 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:11283 t and girth at least g ≥ t + 1. The set of all the graphs of order n, containing no cycles of length ≤ t, and of size ex(n; t), is denoted by EX(n; t) = EX(n; {C₃,C₄, . . . ,Cᵼ }), these graphs are called EX graphs. In 1975, Erdős proposed the problem of determining the extremal numbers ex(n; 4) of a graph of order n 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(29; 6) = 45, also we improve some lower bounds and upper bounds of exᴜ(n; t), for some particular values of n and t.]]> Sat 24 Mar 2018 08:12:43 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:10001 Sat 24 Mar 2018 08:12:22 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:11673 Sat 24 Mar 2018 08:08:43 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:17583 Sat 24 Mar 2018 08:03:58 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:17514 Sat 24 Mar 2018 08:03:50 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:21311 Sat 24 Mar 2018 07:54:39 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:19421 degree/diameter problem, namely, given natural numbers d≥2 and D≥2, find the maximum number _{Nᵇ(d, D)} of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound _Mᵇ(d,D) represents a general upper bound for _Nᵇ(d,D). Bipartite graphs of order _Mᵇ(d,D) are very rare, and determining _Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-Purón and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,−4)-graphs (that is, bipartite graphs of order _Mb(d,D)−4) was carried out. Here we first present some structural properties of bipartite (d,3,−4)-graphs, and later prove that there are no bipartite (7,3,−4)-graphs. This result implies that the known bipartite (7,3,−6)-graph is optimal, and therefore _Nᵇ(7,3)=80. We dub this graph the Hafner–Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,−4)-graph, and the non-existence of bipartite (6,3,−4)-graphs. In addition, we discover at least one new largest known bipartite–and also vertex-transitive–graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for _Nᵇ(11,3).]]> Sat 24 Mar 2018 07:51:59 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:19436 Sat 24 Mar 2018 07:51:58 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:18822 Sat 24 Mar 2018 07:51:08 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:5916 Sat 24 Mar 2018 07:46:46 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:5919 Sat 24 Mar 2018 07:46:46 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:5744 Sat 24 Mar 2018 07:45:03 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:68 Sat 24 Mar 2018 07:42:02 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: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://nova.newcastle.edu.au/vital/access/manager/Repository/uon:28611 Sat 24 Mar 2018 07:38:54 AEDT ]]> https://nova.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://nova.newcastle.edu.au/vital/access/manager/Repository/uon:26777 Sat 24 Mar 2018 07:36:23 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:29677 Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees.]]> Sat 24 Mar 2018 07:32:21 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:27524 Sat 24 Mar 2018 07:28:56 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:28309 1, diameter k > 1 and order N(d,k)=d+d²+...+d^k. So far, their existence has only been showed for k = 2. Their nonexistence has been proved for k = 3, 4 and for d = 2, 3 when k ≥ 3. In this paper, we prove that (4, k) and (5, k)-digraphs with self-repeats do not exist for infinitely many primes k.]]> Sat 24 Mar 2018 07:27:06 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:27495 Sat 24 Mar 2018 07:25:39 AEDT ]]> https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4799 Sat 24 Mar 2018 07:20:40 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:24372 Sat 24 Mar 2018 07:16:18 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:32759 Mon 23 Jul 2018 11:04:20 AEST ]]> 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: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:33348 Fri 19 Oct 2018 12:56:44 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 ]]>