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Please visit our group, The Combinatorial
Mathematics Group, page to get more information about the projects and the
people involved. In the design of large interconnection
networks there is usually a need for each pair of nodes to communicate or to
exchange data efficiently, but it is impractical to directly connect each
pair of nodes. Therefore, the problem of designing networks is concerned with
two constraints: the limitation of the number of connections attached to
every node and also the limitation of the number of intermediate nodes on the
communication route between any two given nodes. The first constraint is modeled by the degree
of a node. The second constraint is modeled by distance parameters
of a graph, such as diameter, radius, or average distance. We are
particularly interested in the diameter parameter. Thus in graph theoretical terms, the problem
becomes The Degree/Diameter Problem For given d and k, find graphs (if the network is
unidirectional) or digraphs (for a bidirectional network) of maximum degree
or outdegree d and diameter at most k with maximum number of
vertices n_{d,k}. A wellknown upper bound for n_{d,k} is the
socalled In general, the degree/diameter problem is
not an easy problem to solve. For instance, the existence of · establishing upper
bound for n_{d,k} by proving the
nonexistence of graphs or digraphs with order "close" to the Moore
bound and · establishing lower
bound for n_{d,k} by constructing
large graphs and digraphs progressed slowly during the
years; the gaps between the two bounds are still enormous, especially for
large values of d and k. An uptodate table of largest known graph
for small values of d and k can be found here. Our contributions towards the solutions of
the problem include: The Neighbourhood Theorem (a powerful
tool to improve nonexistence of graphs with certain properties), the
nonexistence of graphs with degree 4 and order two less than A more complete description of the
degree/diameter problem is provided in ?. Related publications: ·
Mirka Miller, Minh Hoang Nguyen and Rinovia Simanjuntak, Repeats
in graphs and digraphs, Proceedings of EuroComb
2003, to appear. ·
Rinovia Simanjuntak and Mirka Miller, Neighbourhood
Lemma for graphs of order two less than the Moore bound, Technical
Report No. CS02012 (2002) ·
Rinovia Simanjuntak and Mirka Miller, Largest planar digraphs of diameter 2,
Proceedings of Thirteenth Australasian Workshop of Combinatorial
Algorithm, Fraser Island, Australia, 710 July 2001. ·
Rinovia Simanjuntak and Edy Tri Baskoro, The
uniqueness of almost Moore digraph with degree 4 and diameter 2, Proceeding
of SEAMSGMU International Confference 1999 on
Mathematics and its Applications, Jogjakarta,
Indonesia, 2629 July 1999. See also Proc. ITB Vol
32 No 1 (2000) 711.
A labeling (or valuation)
of a graph is any mapping that sends some set of graph elements to a set of
numbers (usually positive or nonnegative integers). If the domain is the
vertexset or the edgeset or both, the labelings
are called respectively vertexlabelings
or edgelabelings or totallabelings. A complete survey of graph labelings,
written by Joseph Gallian, can be downloaded in The Electronic
Journal of Combinatorics. I am working on some special types of graph labelings that are called magic and antimagic graph labelings.
We especially consider the socalled (a,d)antimagic labelings;
and in this one type of graph labelings we have
devised a new terminology for defining such labelings
and found all posible combinations of the labelings within our terminology, studied properties of
the labelings, and constructed labelings
for several families of graphs. Together with Chris Rodger, we also
introduced labelings that are both magic and
distancebased; and we called such labelings the dmagic
labelings, with d denote the desirable
distance in the labelings. With Edy Baskoro, we are working on the application of the magic
labeling to secret sharing scheme, by introducing a set called a critical
set of a particular magic labeling of a graph. Related publications: ·
Mirka Miller, Chris Rodger and Rinovia Simanjuntak, Distance magic labeling of graphs, Australasian
J. of Combinatorics, to appear. ·
Martin Bača, François Bertault,
James MacDougall, Mirka Miller, Rinovia
Simanjuntak, and Slamin, Vertexantimagic total labelings of
graphs, Discussiones Mathematicae Graph Theory, to appear. ·
Slamin, Martin Bača, Yuqing Lin, Mirka Miller, and Rinovia Simanjuntak, Edgemagic
total labelings of wheels, fans and friendship
graphs, Bulletin of ICA, 35 (2002) 8998. ·
Martin Bača, Yuqing Lin, Mirka Miller, Rinovia Simanjuntak, New
constructions of magic and antimagic graph labelings, Util. Math.
60 (2001) 229239. ·
Mirka Miller, Chris Rodger and Rinovia Simanjuntak, On distance magic graphs, Proceedings
of Twelfth Australasian Workshop of Combinatorial Algorithm, Lembang, Indonesia, 1417 July 2001 . ·
YuQing Lin, Mirka Miller, Rinovia Simanjuntak, and Slamin, Magic and antimagic labelings of wheels, Proceedings of Eleventh
Australasian Workshop of Combinatorial Algorithm, Hunter Valley,
Australia, 29 July  1 August 2000. ·
Rinovia Simanjuntak, Mirka Miller, and François Bertault,
Two new (a,d)antimagic
graph labelings, Proceedings of Eleventh
Australasian Workshop of Combinatorial Algorithm, Hunter Valley, Australia,
29 July  1 August 2000. ·
Rinovia Simanjuntak and Mirka Miller, A survey of (a,d)antimagic graph labelings, MIHMI
(Journal of the Indonesian Mathematical Society) Vol. 6 No 5,
Prosiding Konperensi
Nasional Matematika X,
Bandung, Indonesia, 1720 July 2000.



presented in
English and Indonesian @20032006 rino ~ email me last updated: 4
December 2006 