Notes on simplicial rook graphs |
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Authors: | Andries E Brouwer Sebastian M Cioab? Willem H Haemers Jason R Vermette |
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Institution: | 1.Amsterdam,The Netherlands;2.Department of Mathematical Sciences,University of Delaware,Newark,USA;3.Department of Econometrics and Operations Research,Tilburg University,Tilburg,The Netherlands;4.Natural Sciences Division,Missouri Baptist University,Saint Louis,USA |
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Abstract: | The simplicial rook graph \(\mathrm{SR}(m,n)\) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that \(\mathrm{SR}(m,n)\) has integral eigenvalues, and smallest eigenvalue \(s = \max \left( -n, -{m \atopwithdelims ()2}\right) \), and that this graph has a large part of its spectrum in common with the Johnson graph \(J(m+n-1,n)\). We determine the automorphism group and several other properties. |
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