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Maximal Spacing Configurations in Graphs

Published online by Cambridge University Press:  01 December 1997

Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE, UK; (e-mail:
Department of Mathematical Statistics and Operational Research, University of Exeter, North Park Road, Exeter EX4 4QE, UK; (e-mail:


Let G=(V, E) be a simple connected graph of order [mid ]V[mid ]=n[ges ]2 and minimum degree δ, and let 2[les ]s[les ]n. We define two parameters, the s-average distance μs(G) and the s-average nearest neighbour distance Λs(G), with respect to each of which V contains an extremal subset X of order s with vertices ‘as spread out as possible’ in G. We compute the exact values of both parameters when G is the cycle Cn, and show how to obtain the corresponding optimal arrangements of X. Sharp upper and lower bounds are then established for Λs(G), as functions of s, n and δ, and the extremal graphs described.

Research Article
1997 Cambridge University Press

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