https://ogma.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:45210 v with respect to a resolving set W is expressed as a multiset of distances between v and all vertices in W, including their multiplicities. The multiset dimension is defined to be the minimum cardinality of the resolving set. Clearly, this is at least the metric dimension of a graph. In this paper, we study the properties of the multiset dimension of graphs.]]> Wed 26 Oct 2022 17:48:21 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:53324 Wed 22 Nov 2023 10:05:34 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8770 Wed 11 Apr 2018 16:37:45 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8767 Wed 11 Apr 2018 15:52:51 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8766 Wed 11 Apr 2018 14:53:35 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:23527 Wed 11 Apr 2018 13:29:15 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7829 Wed 11 Apr 2018 11:18:43 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:5502 Wed 11 Apr 2018 10:35:09 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30604 Wed 11 Apr 2018 10:30:31 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:4975 Wed 11 Apr 2018 09:35:56 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:46035 Wed 09 Nov 2022 15:31:03 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32179 k_n has many good properties. In this paper, we prove that (1) the connectivity of XQ^k_n is 4n; (2) the nature connectivity of XQ^k_n is 8n − 4; (3) the nature diagnosability of XQ^k_n under the PMC model and MM∗ model is 8n − 3 for n ≥ 2.]]> Wed 09 May 2018 14:58:09 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30313 Wed 09 Feb 2022 15:59:08 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:42921 Wed 07 Sep 2022 15:00:16 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:43995 Wed 05 Oct 2022 14:50:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:55519 Wed 05 Jun 2024 08:40:48 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:50718 Wed 02 Aug 2023 18:30:37 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:50699 Wed 02 Aug 2023 11:58:45 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:52917 Wed 01 Nov 2023 09:29:33 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:54054 Tue 30 Jan 2024 13:58:47 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:55407 Tue 28 May 2024 11:05:37 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:50434 Tue 25 Jul 2023 19:08:37 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:16088 Tue 24 Aug 2021 14:27:38 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:16087 Tue 24 Aug 2021 14:27:30 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:16095 Tue 24 Aug 2021 14:26:03 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:42426 Tue 23 Aug 2022 09:44:16 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:45497 Tue 19 Mar 2024 15:42:12 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:46304 Tue 15 Nov 2022 10:07:48 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:38837 Tue 15 Feb 2022 14:02:29 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:50117 a be two positive integers. We say that G has all [a, b]-factors if G has an h-factor for every h: V→ N such that a≤ h(v) ≤ b for every v∈ V and ∑v∈Vh(v)≡0(mod2). A spanning subgraph F of G is called an (a, b)-parity factor, if dF(v) ≡ a≡ b (mod 2) and a≤ dF(v) ≤ b for all v∈ V. In this paper, we have developed sufficient conditions for the existence of all [a, b]-factors and (a, b)-parity factors of G in terms of the independence number and connectivity of G. This work extended an earlier result of Nishimura (J Graph Theory 13: 63–69, 1989). Furthermore, we show that these results are best possible in some cases.]]> Tue 11 Jul 2023 15:47:27 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:37306 Tue 10 Nov 2020 15:50:27 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:45951 Tue 08 Nov 2022 10:14:14 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:45942 Tue 08 Nov 2022 10:07:05 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:33000 k(G), is defined as the cardinality of a minimum k-restricted edge cut. A connected graph G is said to be λ_k-connected if G has a k-restricted edge cut. Let ξ_k(G) = min{|[X, ̅X ]| : |X| = k, G[X] is connected}, where ̅X = V (G)X. A graph G is said to be maximally k-restricted edge connected if λ_k(G) = ξ_k(G). In this paper we show that if G is a λ₄-connected graph with λ₄(G) ≤ ξ₄(G) and the girth satisfies g(G) ≥ 8, and there do not exist six vertices u₁, u₂, u₃, v₁, v₂ and v₃ in G such that the distance d(u_i, v_j) ≥ 3, (1 ≤ i, j ≤ 3), then G is maximally 4-restricted edge connected.]]> Tue 03 Sep 2019 18:17:58 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32060 Thu 26 Apr 2018 15:20:50 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32057 n-dimensional bubble-sort star graph BS_n has many good properties. In this paper, we prove that the nature diagnosability of BS_n is 4n - 7 under the PMC model for n ≥ 4, the nature diagnosability of BS_n is n ≥ 4 under the MM* model for n ≥ 5 .]]> Thu 26 Apr 2018 14:24:55 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:379 Thu 25 Jul 2013 09:10:05 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:51055 Thu 17 Aug 2023 10:22:44 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30086 Thu 12 Apr 2018 13:44:01 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:38168 Thu 05 Aug 2021 15:22:40 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:9241 Sat 24 Mar 2018 11:12:48 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31776 Sat 24 Mar 2018 08:44:16 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31779 Sat 24 Mar 2018 08:44:15 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31313 n,k,m be three positive integers such that k=⎣(n - 1)/2⎦ and m≤k, we show that every 2k-regular m-edge-connected graph with 2n vertices contains at least m edge-disjoint perfect matchings, and the condition on edge connectivity is sharp.]]> Sat 24 Mar 2018 08:43:31 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31787 Sat 24 Mar 2018 08:42:51 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8000 Sat 24 Mar 2018 08:42:34 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8001 Sat 24 Mar 2018 08:42:34 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7866 Sat 24 Mar 2018 08:38:57 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8162 Sat 24 Mar 2018 08:36:06 AEDT ]]> https://ogma.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://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8038 Sat 24 Mar 2018 08:35:04 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:10135 Sat 24 Mar 2018 08:13:43 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:11199 Sat 24 Mar 2018 08:13:36 AEDT ]]> https://ogma.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://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:11640 Sat 24 Mar 2018 08:10:41 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:10371 Sat 24 Mar 2018 08:08:50 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:11783 Sat 24 Mar 2018 08:07:34 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19488 Sat 24 Mar 2018 08:02:21 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:17310 Sat 24 Mar 2018 08:01:51 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:17650 Sat 24 Mar 2018 07:57:18 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:16303 Sat 24 Mar 2018 07:53:18 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:16304 Sat 24 Mar 2018 07:53:18 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:20172 Sat 24 Mar 2018 07:51:42 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:5919 Sat 24 Mar 2018 07:46:46 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:299 Sat 24 Mar 2018 07:42:24 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:4799 Sat 24 Mar 2018 07:20:40 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:22970 Sat 24 Mar 2018 07:15:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:22971 n with connectivity κ(G)≤k and minimum degree δ(G)≥k. We show that among the graphs in this family, the maximum spectral radius is obtained uniquely at K_k+(K_δ−k+1∪K_n−δ−1). Another family of the graphs we study is the family of bipartite graphs with order n and connectivity k. We show that among the graphs in this family the maximum spectral radius is obtained at a graph modified from K_{⌊n/2⌋,n−1−⌊n/2⌋}. The third family of graphs we have studied is the family of graphs with order n, connectivity k and independence number r. We determine the graphs in this family that have the maximum spectral radius.]]> Sat 24 Mar 2018 07:15:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:22972 Discrete Math., 101, 33–37 (1992)) has extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has an even factor without isolated vertices. Let me_e > 0 be even and m_o > 0 be odd. In this paper, we prove that every m_e-edge-connected graph with minimum degree at least m_e + 1 contains an even factor with minimum degree at least m_e and every (m_o + 1)-edge-connected graph contains an odd factor with minimum degree at least m_o, which further extends Fleischner’s result. Moreover, we show that our results are best possible.]]> Sat 24 Mar 2018 07:15:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:24872 n generated by the transposition tree Γ_n has many good properties. In this paper, we give that the 2-good-neighbor diagnosability of CΓ_n under the PMC model and MM⁎ model is g(n−2)−1, where n≥4and g is the girth of CΓ_n.]]> Sat 24 Mar 2018 07:11:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:48190 Sat 11 Mar 2023 12:29:43 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32178 et al. proposed the g-extra diagnosability of G, which restrains that every component of G - S has at least (g + 1) vertices. The locally twisted cube LTQ_n is applied widely. In this paper, we show that LTQ_n is tightly (4n-9) super 3-extra connected for n≥6 and the 3-extra diagnosability of LTQ_n under the PMC model and MM* model is 4n - 6 for n≥5 and n≥7, respectively.]]> Mon 23 Sep 2019 12:57:15 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32495 G be a graph with vertex set ^V(G) and edge set ^E(G). We consider the problem of orienting the edges of a complete bipartite graph ^K_n,n so only two different in-degrees ^a and ^b occur. An obvious necessary condition for orienting the edges of ^G so that only two in-degrees ^a and ^b occur, is that there exist positive integers ^s and ^t satisfying ^s+t=|V(G)| and ^as+bt=|V(G)|. In this paper, we show that the necessary condition is also sufficient for a complete bipartite graph ^K_n,n. Furthermore, we give the algorithms of orientations with only two in-degrees of ^K_n,n.]]> Mon 23 Sep 2019 11:31:42 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:53271 Mon 20 Nov 2023 12:56:42 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:44593 Mon 17 Oct 2022 14:46:36 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:52523 Mon 16 Oct 2023 10:45:00 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:39139 Mon 16 May 2022 15:13:41 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:52354 Mon 09 Oct 2023 15:04:38 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:45492 Fri 28 Oct 2022 15:46:09 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:38319 Fri 27 Aug 2021 10:59:38 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32090 Fri 27 Apr 2018 15:50:51 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:34717 Fri 26 Apr 2019 15:52:14 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:49355 Fri 12 May 2023 12:28:02 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:41738 Fri 12 Aug 2022 11:07:32 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:46942 Fri 10 Nov 2023 15:25:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:40371 Fri 08 Jul 2022 16:16:33 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:55590 Fri 07 Jun 2024 12:09:19 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30752 G and the graph G + K₁. Finally, we determine the distance irregularity strength of m-book graphs B_m.]]> Fri 06 Oct 2023 15:48:06 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:38177 Fri 06 Aug 2021 14:49:41 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:55257 Fri 03 May 2024 15:23:38 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:46844 Fri 02 Dec 2022 08:24:22 AEDT ]]>