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Abstract
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For any positive definite rational quadratic form
of
variables let
denote the graph
with vertices
and
connected
if and only if
.
This notion generalises standard Euclidean distance graphs. In this article we study
these graphs and show how to find the exact value of clique number of the
.
We also prove rational analogue of the Beckman–Quarles theorem that any unit-preserving
bijection of
onto itself is an isometry.
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Keywords
Euclidean distance graph, rational points, quadratic form,
clique, regular simplex, Beckman–Quarles theorem
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Mathematical Subject Classification
Primary: 05C60, 05C69, 52C35
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Milestones
Received: 3 March 2023
Revised: 4 April 2023
Accepted: 18 April 2023
Published: 4 June 2023
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© 2023 MSP (Mathematical Sciences
Publishers). |
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