Abstract
We discuss the range of values for the integrity of a graphs G(n, k) where G(n, k) denotes a simple graph with n vertices and k edges. Let I max(n, k) and I min(n, k) be the maximal and minimal value for the integrity of all possible G(n, k) graphs and let the difference be D(n, k) = I max(n, k) − I min(n, k). In this paper we give some exact values and several lower bounds of D(n, k) for various values of n and k. For some special values of n and for s < n 1/4 we construct examples of graphs G n = G n (n, n + s) with a maximal integrity of I(G n ) = I(C n ) + s where C n is the cycle with n vertices. We show that for k = n 2/6 the value of D(n, n 2/6) is at least \({\frac{\sqrt{6}-1}{3}n}\) for large n.
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Claus Ernst was supported by NSF Grant #DMS-0310562.
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Atici, M., Ernst, C. On the Range of Possible Integrities of Graphs G(n, k). Graphs and Combinatorics 27, 475–485 (2011). https://doi.org/10.1007/s00373-010-0990-1
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DOI: https://doi.org/10.1007/s00373-010-0990-1