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Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders
Joint Authors
Al-Addasi, Salah
Al-Ezeh, Hasan
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2008, Issue 2008 (31 Dec. 2008), pp.1-12, 12 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2007-12-10
Country of Publication
Egypt
No. of Pages
12
Main Subjects
Abstract EN
We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets.
For a bipartite diametrical graph of diameter 4 and partite sets U and W, where 2m=|U|≤|W|, we prove that 2m is a sharp upper bound of |W| and construct an S-graph G(2m,2m) in which this upper bound is attained, this graph can be viewed as a generalization of the Rhombic Dodecahedron.
Then we show that for any m≥2, the graph G(2m,2m) is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets of cardinalities 2m and 2m, and hence in particular, for m=3, the graph G(6,8) which is just the Rhombic Dodecahedron is the unique (up to isomorphism) bipartite diametrical graph of such a diameter and cardinalities of partite sets.
Thus we complete a characterization of S-graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6.
We prove that the neighborhoods of vertices of the larger partite set of G(2m,2m) form a matroid whose basis graph is the hypercube Qm.
We prove that any S-graph of diameter 4 is bipartite self complementary, thus in particular G(2m,2m).
Finally, we study some additional properties of G(2m,2m) concerning the order of its automorphism group, girth, domination number, and when being Eulerian.
American Psychological Association (APA)
Al-Addasi, Salah& Al-Ezeh, Hasan. 2007. Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders. International Journal of Mathematics and Mathematical Sciences،Vol. 2008, no. 2008, pp.1-12.
https://search.emarefa.net/detail/BIM-987892
Modern Language Association (MLA)
Al-Addasi, Salah& Al-Ezeh, Hasan. Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders. International Journal of Mathematics and Mathematical Sciences No. 2008 (2008), pp.1-12.
https://search.emarefa.net/detail/BIM-987892
American Medical Association (AMA)
Al-Addasi, Salah& Al-Ezeh, Hasan. Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders. International Journal of Mathematics and Mathematical Sciences. 2007. Vol. 2008, no. 2008, pp.1-12.
https://search.emarefa.net/detail/BIM-987892
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-987892