On the Existence of S-graphs

ECSU Author/Contributor (non-ECSU co-authors, if there are any, appear on document)
Julian A. D. Allagan , Associate Professor (Creator)
Elizabeth City State University (ECSU )
Web Site: https://www.ecsu.edu/academics/library/index.html

Abstract: We answer in the affirmative a question posed by S. Al-Addasi and H. Al-Ezeh in [Int. J. Math. Math. Sci. 2008, Article ID 468583, 11 p. (2008; Zbl 1161.05316)] on the existence of symmetric diametrical bipartite graphs of diameter 4. Bipartite symmetric diametrical graphs are calls S-graphs by some authors and diametrical graphs have also been studied by other authors using different terminology, such as self-centered unique eccentric point graphs. We include a brief survey of some of this literature and advertise that the existence question was answered by A. Berman and A. Kotzig [Ann. Discrete Math. 8, 37–42 (1980; Zbl 0446.05025)], along with a study of different isomorphism classes of these graphs using a (1,-1)-matrix representation which includes the well-known Hadamard matrices. Our presentation focuses on a neighborhood characterization of S-graphs and we conclude our survey with a beautiful version of this characterization known to T. N. Janakiraman [Discrete Math. 126, No. 1–3, 411–414 (1994; Zbl 0792.05117)]

Additional Information

Language: English
Date: 2012
cartesian product, geodesic, S??graphs, symmetric diametrical, neighborhood characterization

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