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14Miller, Mirka
7Lin, Yuqing
5Balbuena, Camino
5Sugeng, Kiki Ariyanti
3Marcote, Xavier
3Ryan, Joe
2Barker, Ewan
2Baskoro, Edy
2Dzalilov, Zari
2Gimbert, Joan
1Bagirov, Adil
1Cholily, Yus Mochamad
1Das, K. C.
1Fernau, Henning
1Hart, Sol
1Jiang, T.
1Jin, Ruoming
1Lee, Victor
1Lopez, Nacho
1Marti, J. Gomez

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120101 Pure Mathematics
7Numerical methods
4Connectivity
4Moore bound
4Number theory
4Problem solving
4Set theory
4Theorem proving
3Integer programming
2(k,g)-cage
20802 Computation Theory and Mathematics
2Digraphs
2Dynamic systems theory
2Finite difference method
2Magic labeling
2Optimisation
2Path analysis
2Simulation
2Telecommunication network failures

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A highway-centric labeling approach for answering distance queries on large sparse graphs

- Jin, Ruoming, Ruan, Ning, Xiang, Yang, Lee, Victor

**Authors:**Jin, Ruoming , Ruan, Ning , Xiang, Yang , Lee, Victor**Date:**2012**Type:**Text , Conference paper**Relation:**ACM SIGMOD International Conference on Management of Data, 2012 p. 445-456**Full Text:**false**Reviewed:****Description:**The distance query, which asks the length of the shortest path from a vertex u to another vertex v, has applications ranging from link analysis, semantic web and other ontology processing, to social network operations. Here, we propose a novel labeling scheme, referred to as Highway-Centric Labeling, for answering distance queries in a large sparse graph. It empowers the distance labeling with a highway structure and leverages a novel bipartite set cover framework/algorithm. Highway-centric labeling provides better labeling size than the state-of-the-art 2-hop labeling, theoretically and empirically. It also offers both exact distance and approximate distance with bounded accuracy. A detailed experimental evaluation on both synthetic and real datasets demonstrates that highway-centric labeling can outperform the state-of-the-art distance computation approaches in terms of both index size and query time. © 2012 ACM.**Description:**Proceedings of the ACM SIGMOD International Conference on Management of Data

A sum labelling for the generalised friendship graph

- Fernau, Henning, Ryan, Joe, Sugeng, Kiki Ariyanti

**Authors:**Fernau, Henning , Ryan, Joe , Sugeng, Kiki Ariyanti**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 5-6 (2008), p. 734-740**Full Text:**false**Reviewed:****Description:**We provide an optimal sum labelling scheme for the generalised friendship graph, also known as the flower (a symmetric collection of cycles meeting at a common vertex) and show that its sum number is 2. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

Enumerations of vertex orders of almost Moore digraphs with selfrepeats

- Baskoro, Edy, Cholily, Yus Mochamad, Miller, Mirka

**Authors:**Baskoro, Edy , Cholily, Yus Mochamad , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 1 (2008), p. 123-128**Full Text:**false**Reviewed:****Description:**An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r (u) = v, such that there are two walks of length ≤ k from u to v. The smallest positive integer p such that the composition rp (u) = u is called the order of u. If the order of u is 1 then u is called a selfrepeat. It is known that if G is an almost Moore digraph of diameter k ≥ 3 then G contains exactly k selfrepeats or none. In this paper, we propose an exact formula for the number of all vertex orders in an almost Moore digraph G containing selfrepeats, based on the vertex orders of the out-neighbours of any selfrepeat vertex. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

Graphs of order two less than the Moore bound

- Miller, Mirka, Simanjuntak, Rinovia

**Authors:**Miller, Mirka , Simanjuntak, Rinovia**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 13 (2008), p. 2810-2821**Full Text:**false**Reviewed:****Description:**The Moore bound for a directed graph of maximum out-degree d and diameter k is Md,k=1+d+d2++dk. It is known that digraphs of order Md,k (Moore digraphs) do not exist for d>1 and k>1. Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k is . Undirected Moore graphs only exist in a small number of cases. Mixed (or partially directed) Moore graphs generalize both undirected and directed Moore graphs. In this paper, we shall show that all known mixed Moore graphs of diameter k=2 are unique and that mixed Moore graphs of diameter k3 do not exist.**Description:**C1

HSAGA and its application for the construction of near-Moore digraphs

- Tang, Jianmin, Miller, Mirka, Lin, Yuqing

**Authors:**Tang, Jianmin , Miller, Mirka , Lin, Yuqing**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Discrete Algorithms Vol. 6, no. 1 (2008), p. 73-84**Full Text:**false**Reviewed:****Description:**The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. This paper deals with directed graphs. General upper bounds, called Moore bounds, exist for the largest possible order of such digraphs of maximum degree d and given diameter k. It is known that simulated annealing and genetic algorithm are effective techniques to identify global optimal solutions. This paper describes our attempt to build a Hybrid Simulated Annealing and Genetic Algorithm (HSAGA) that can be used to construct large digraphs. We present our new results obtained by HSAGA, as well as several related open problems. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

On consecutive edge magic total labeling of graphs

- Sugeng, Kiki Ariyanti, Miller, Mirka

**Authors:**Sugeng, Kiki Ariyanti , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Discrete Algorithms Vol. 6, no. 1 (2008), p. 59-65**Full Text:**false**Reviewed:****Description:**Let G = (V, E) be a finite (non-empty) graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection**Description:**C1

On the connectivity of (k, g)-cages of even girth

- Lin, Yuqing, Balbuena, Camino, Marcote, Xavier, Miller, Mirka

**Authors:**Lin, Yuqing , Balbuena, Camino , Marcote, Xavier , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 15 (2008), p. 3249-3256**Full Text:**false**Reviewed:****Description:**A (k,g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)-cages for g even. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

A lower bound on the order of regular graphs with given girth pair

- Balbuena, Camino, Jiang, T., Lin, Yuqing, Marcote, Xavier, Miller, Mirka

**Authors:**Balbuena, Camino , Jiang, T. , Lin, Yuqing , Marcote, Xavier , Miller, Mirka**Date:**2007**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 55, no. 2 (2007), p. 153-163**Full Text:**false**Reviewed:****Description:**The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovács [Regular graphs with given girth pair, J Graph Theory 7 (1983), 209-218]. A (**Description:**C1**Description:**2003004727

- Nguyen, Minh Hoang, Miller, Mirka, Gimbert, Joan

**Authors:**Nguyen, Minh Hoang , Miller, Mirka , Gimbert, Joan**Date:**2007**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 307, no. 7-8 (2007), p. 964-970**Full Text:**false**Reviewed:****Description:**The Moore bound for a directed graph of maximum out-degree d and diameter k is M**Description:**C1**Description:**2003005024

Visual tools for analysing evolution, emergence, and error in data streams

- Hart, Sol, Yearwood, John, Bagirov, Adil

**Authors:**Hart, Sol , Yearwood, John , Bagirov, Adil**Date:**2007**Type:**Text , Conference paper**Relation:**Paper presented at 6th IEEE/ACIS International Conference on Computer and Information Science, ICIS 2007, Melbourne, Victoria : 11th-13th July 2007 p. 987-992**Full Text:****Description:**The relatively new field of stream mining has necessitated the development of robust drift-aware algorithms that provide accurate, real time, data handling capabilities. Tools are needed to assess and diagnose important trends and investigate drift evolution parameters. In this paper, we present two new and novel visualisation techniques, Pixie and Luna graphs, which incorporate salient group statistics coupled with intuitive visual representations of multidimensional groupings over time. Through the novel representations presented here, spatial interactions between temporal divisions can be diagnosed and overall distribution patterns identified. It provides a means of evaluating in non-constrained capacity, commonly constrained evolutionary problems.**Description:**2003005432

**Authors:**Hart, Sol , Yearwood, John , Bagirov, Adil**Date:**2007**Type:**Text , Conference paper**Relation:**Paper presented at 6th IEEE/ACIS International Conference on Computer and Information Science, ICIS 2007, Melbourne, Victoria : 11th-13th July 2007 p. 987-992**Full Text:****Description:**The relatively new field of stream mining has necessitated the development of robust drift-aware algorithms that provide accurate, real time, data handling capabilities. Tools are needed to assess and diagnose important trends and investigate drift evolution parameters. In this paper, we present two new and novel visualisation techniques, Pixie and Luna graphs, which incorporate salient group statistics coupled with intuitive visual representations of multidimensional groupings over time. Through the novel representations presented here, spatial interactions between temporal divisions can be diagnosed and overall distribution patterns identified. It provides a means of evaluating in non-constrained capacity, commonly constrained evolutionary problems.**Description:**2003005432

All (k;g)-cages are edge-superconnected

- Lin, Yuqing, Miller, Mirka, Balbuena, Camino, Marcote, Xavier

**Authors:**Lin, Yuqing , Miller, Mirka , Balbuena, Camino , Marcote, Xavier**Date:**2006**Type:**Text , Journal article**Relation:**Networks Vol. 47, no. 2 (2006), p. 102-110**Full Text:**false**Reviewed:****Description:**A (k;g)-cage is k-regular graph with girth g and with the least possible number of vertices. In this article we prove that (k;g)-cages are edge-superconnected if g is even. Earlier, Marcote and Balbuena proved that (k;g)-cages are edge-superconnected if g is odd [Networks 43 (2004), 54-59]. Combining our results, we conclude that all (k;g)-cages are edge-superconnected. © 2005 Wiley Periodicals, Inc.**Description:**C1**Description:**2003001830

Characterization of eccentric digraphs

- Gimbert, Joan, Lopez, Nacho, Miller, Mirka, Ryan, Joe

**Authors:**Gimbert, Joan , Lopez, Nacho , Miller, Mirka , Ryan, Joe**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 2 (2006), p. 210-219**Full Text:**false**Reviewed:****Description:**The eccentric digraph ED(G) of a digraph G represents the binary relation, defined on the vertex set of G, of being 'eccentric'; that is, there is an arc from u to v in ED(G) if and only if v is at maximum distance from u in G. A digraph G is said to be eccentric if there exists a digraph H such that G=ED(H). This paper is devoted to the study of the following two questions: what digraphs are eccentric and when the relation of being eccentric is symmetric. We present a characterization of eccentric digraphs, which in the undirected case says that a graph G is eccentric iff its complement graph G is either self-centered of radius two or it is the union of complete graphs. As a consequence, we obtain that all trees except those with diameter 3 are eccentric digraphs. We also determine when ED(G) is symmetric in the cases when G is a graph or a digraph that is not strongly connected. Crown Copyright © 2006 Published by Elsevier B.V. All rights reserved.**Description:**C1**Description:**2003001601

- Balbuena, Camino, Barker, Ewan, Lin, Yuqing, Miller, Mirka, Sugeng, Kiki Ariyanti

**Authors:**Balbuena, Camino , Barker, Ewan , Lin, Yuqing , Miller, Mirka , Sugeng, Kiki Ariyanti**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 16 (2006), p. 1817-1829**Full Text:**false**Reviewed:****Description:**Let G be a graph of order n and size e. A vertex-magic total labeling is an assignment of the integers 1, 2, ..., n + e to the vertices and the edges of G, so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant, called the magic number of G. Such a labeling is a-vertex consecutive magic if the set of the labels of the vertices is { a + 1, a + 2, ..., a + n }, and is b-edge consecutive magic if the set of labels of the edges is { b + 1, b + 2, ..., b + e }. In this paper we prove that if an a-vertex consecutive magic graph has isolated vertices then the order and the size satisfy (n - 1)**Description:**C1**Description:**2003001604

On the degrees of a strongly vertex-magic graph

- Balbuena, Camino, Barker, Ewan, Das, K. C., Lin, Yuqing, Miller, Mirka, Ryan, Joe, Slamin,, Sugeng, Kiki Ariyanti, Tkac, M.

**Authors:**Balbuena, Camino , Barker, Ewan , Das, K. C. , Lin, Yuqing , Miller, Mirka , Ryan, Joe , Slamin, , Sugeng, Kiki Ariyanti , Tkac, M.**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 6 (2006), p. 539-551**Full Text:**false**Reviewed:****Description:**Let G=(V,E) be a finite graph, where |V|=n≥2 and |E|=e≥1. A vertex-magic total labeling is a bijection λ from V∪E to the set of consecutive integers {1,2,...,n+e} with the property that for every v∈V, λ(v)+∑w∈N(v)λ(vw)=h for some constant h. Such a labeling is strong if λ(V)={1,2,...,n}. In this paper, we prove first that the minimum degree of a strongly vertex-magic graph is at least two. Next, we show that if 2e≥10n2-6n+1, then the minimum degree of a strongly vertex-magic graph is at least three. Further, we obtain upper and lower bounds of any vertex degree in terms of n and e. As a consequence we show that a strongly vertex-magic graph is maximally edge-connected and hamiltonian if the number of edges is large enough. Finally, we prove that semi-regular bipartite graphs are not strongly vertex-magic graphs, and we provide strongly vertex-magic total labeling of certain families of circulant graphs. © 2006 Elsevier B.V. All rights reserved**Description:**C1**Description:**2003001603

Two new families of large compound graphs

- Marti, J. Gomez, Miller, Mirka

**Authors:**Marti, J. Gomez , Miller, Mirka**Date:**2006**Type:**Text , Journal article**Relation:**Networks Vol. 47, no. 3 (2006), p. 140-146**Full Text:**false**Reviewed:****Description:**A question of special interest in graph theory is the design of large graphs. Specifically, we want to find constructions of graphs with order as large as possible for a given degree A and diameter D. Two generalizations of two large compound graphs are proposed in this article. Three particular cases of these families of graphs presented here allow us to improve the order for the entries (15, 7), (13, 10), and (15, 10) in the table of the largest known (Δ, D)-graphs. © 2006 Wiley Periodicals, Inc.**Description:**C1**Description:**2003001599

All (k;g)-cages are k-edge-connected

- Lin, Yuqing, Miller, Mirka, Rodger, Chris

**Authors:**Lin, Yuqing , Miller, Mirka , Rodger, Chris**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 48, no. 3 (2005), p. 219-227**Full Text:**false**Reviewed:****Description:**A (k;g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)-cages are k-edge-connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)-cages are k-edge-connected if g is odd. Combining our results, we conclude that the (k;g)-cages are k-edge-connected. © 2005 wiley Periodicals, Inc.**Description:**C1

Complete characterization of almost moore digraphs of degree three

- Baskoro, Edy, Miller, Mirka, Siran, Jozef, Sutton, Martin

**Authors:**Baskoro, Edy , Miller, Mirka , Siran, Jozef , Sutton, Martin**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 48, no. 2 (2005), p. 112-126**Full Text:**false**Reviewed:****Description:**It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ≥ 3 which miss the Moore bound by one do not exist. © 2004 Wiley Periodicals, Inc.**Description:**C1**Description:**2003000904

Magic and antimagic labeling of graphs

**Authors:**Sugeng, Kiki Ariyanti**Date:**2005**Type:**Text , Thesis , PhD**Full Text:****Description:**"A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."**Description:**Doctor of Philosophy

**Authors:**Sugeng, Kiki Ariyanti**Date:**2005**Type:**Text , Thesis , PhD**Full Text:****Description:**"A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."**Description:**Doctor of Philosophy

A framework for monitoring progress and planning teaching towards the effective use of computer algebra systems

**Authors:**Pierce, Robyn , Stacey, Kaye**Date:**2004**Type:**Text , Journal article**Relation:**International Journal of Computers for Mathematical Learning Vol. 9, no. 1 (2004), p. 59-93**Full Text:****Reviewed:****Description:**This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.**Description:**C1**Description:**2003000923

**Authors:**Pierce, Robyn , Stacey, Kaye**Date:**2004**Type:**Text , Journal article**Relation:**International Journal of Computers for Mathematical Learning Vol. 9, no. 1 (2004), p. 59-93**Full Text:****Reviewed:****Description:**This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.**Description:**C1**Description:**2003000923

- Patel, Keyurkumar, Dzalilov, Zari

**Authors:**Patel, Keyurkumar , Dzalilov, Zari**Date:**2004**Type:**Text , Conference paper**Relation:**Paper presented at 4th IASTED International Conference on Modelling, Simulation, and Optimisation, Marbella, Spain : 17-19th August, 2004 p. 219-244**Full Text:**false**Reviewed:****Description:**For the last quarter of a century understanding of the nature of telecommunication network traffic has been considered as an important research topic. Any well designed recovery strategy has to take into account the different resilience requirements of the single traffic flows in order to avoid excessive usage of bandwidth for standby links. Faced with multiple recovery options, an internet service provider (ISP) must decide which flows to protect to what extent against networks. Traditional techniques and models used to determine the availability and failure rates of telecommunications networks are based on classic failure models such as Mean-time between failure and Mean-time between service outage predictors. Network failures occur for many different reasons and occur in many different forms. These classic models only assume that the failure is caused by a hardware component of the network. With the widespread deployment of Internet technologies other factors that cause or contribute to failure in a telecommunications network must be explored. Two additional failure modes to existing published failure models, failure from Denial of Service attacks, and failures due to catastrophic events have been identified and defined along with an initial outline of a generalized prediction model based on Dynamic System Theory. Discussion is done on the effects of failures and survivability issues in network failures and how to overcome failures in IT based telecommunication network**Description:**E1**Description:**2003002567

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