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INCIDENCE SEMIRINGS OF GRAPHS AND VISIBLE BASES

Part of: Generalizations Multivariate analysis

Published online by Cambridge University Press:  12 September 2013

J. ABAWAJY ,
A. V. KELAREV ,
M. MILLER  and
J. RYAN
J. ABAWAJY
Affiliation:
School of Information Technology, Deakin University, 221 Burwood, Melbourne, Victoria 3125, Australia email [email protected]
A. V. KELAREV*
Affiliation:
School of Electrical Engineering and Computer Science, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia email [email protected]
M. MILLER
Affiliation:
CARMA Priority Research Centre, School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia email [email protected] Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic email [email protected]
J. RYAN
Affiliation:
School of Electrical Engineering and Computer Science, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia email [email protected]
*
✉[email protected]
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Abstract

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We consider the incidence semirings of graphs and prove that every incidence semiring has convenient visible bases for its right ideals and for its left ideals, and that these visible bases can be used to determine the weights of all right ideals that have maximum weight and all left ideals that have maximum weight.

Keywords

incidence semiringsright idealsbalanced graphs

MSC classification

Secondary: 16Y60: Semirings 62H30: Classification and discrimination; cluster analysis
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 451 - 459
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Abawajy, J. and Kelarev, A. V., ‘Classification systems based on combinatorial semigroups’, Semigroup Forum 86 (3) (2013), 603–612.CrossRefGoogle Scholar
Abawajy, J., Kelarev, A. V. and Zeleznikow, J., ‘Optimization of classification and clustering systems based on Munn semirings’, Semigroup Forum, doi:10.1007/s00233-013-9488-5.CrossRefGoogle Scholar
Andruszkiewicz, R. R. and Sobolewska, M., ‘On the stabilisation of one-sided Kurosh’s chains’, Bull. Aust. Math. Soc. 86 (2012), 473–480.CrossRefGoogle Scholar
Gao, D. Y., Kelarev, A. V. and Yearwood, J. L., ‘Optimization of matrix semirings for classification systems’, Bull. Aust. Math. Soc. 84 (2011), 492–503.CrossRefGoogle Scholar
Golan, J. S., Semirings and Their Applications (Kluwer Academic Publishers, Dordrecht, 1999).CrossRefGoogle Scholar
Kelarev, A. V., ‘On classical Krull dimension of group-graded rings’, Bull. Aust. Math. Soc. 55 (1997), 255–259.CrossRefGoogle Scholar
Kelarev, A. V., ‘On undirected Cayley graphs’, Aust. J. Combin. 25 (2002), 73–78.Google Scholar
Kelarev, A. V., Ring Constructions and Applications (World Scientific, River Edge, NJ, 2002).Google Scholar
Kelarev, A. V., Graph Algebras and Automata (Marcel Dekker, New York, 2003).CrossRefGoogle Scholar
Kelarev, A. V., ‘Labelled Cayley graphs and minimal automata’, Aust. J. Combin. 30 (2004), 95–101.Google Scholar
Kelarev, A. V. and Passman, D. S., ‘A description of incidence rings of group automata’, Contemp. Math. 456 (2008), 27–33.CrossRefGoogle Scholar
Kelarev, A., Ryan, J. and Yearwood, J., ‘Cayley graphs as classifiers for data mining: the influence of asymmetries’, Discrete Math. 309 (17) (2009), 5360–5369.Google Scholar
Kelarev, A. V., Yearwood, J. L. and Vamplew, P. W., ‘A polynomial ring construction for classification of data’, Bull. Aust. Math. Soc. 79 (2009), 213–225.Google Scholar
Spiegel, E. and O’Donnell, C. J., Incidence Algebras (Marcel Dekker, New York, 1997).Google Scholar