Des. Codes Cryptogr. https://doi.org/10.1007/s10623-018-0498-x The vertex-isoperimetric number of the incidence and non-incidence graphs of unitals 1 2 Alice M. W. Hui · Muhammad Adib Surani · Sanming Zhou Received: 1 December 2017 / Revised: 13 May 2018 / Accepted: 23 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We derive upper and lower bounds for the vertex-isoperimetric number of the incidence graphs of unitals and determine its order of magnitude. In the case when a unital contains sufﬁciently large arcs, these bounds agree and give rise to the precise value of this parameter. In particular, we obtain the exact value of the vertex-isoperimetric number of the incidence graphs of classical unitals and a certain subfamily of BM-unitals. In the case when the maximum size of arcs in the unital is relatively small, we obtain an upper bound for this parameter in terms of the vertex-isoperimetric number of the incidence graph. We also determine the exact value of the vertex-isoperimetric number of the non-incidence graph of any unital. Keywords Vertex-isoperimetric number · Unital · Incidence graph Mathematics Subject Classiﬁcation 05C40 · 05B25 1 Introduction A unital is a 2-(n + 1, n + 1, 1)
Designs, Codes and Cryptography – Springer Journals
Published: May 31, 2018
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