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Optimal values for ∆. n ∆ n ∆ n ∆ n ∆ 12 7 25 13 38 18 51 23 13 8 26 13 39 18 52 24 14 8 27 13 40 19 53 24 15 9 28 14 41 19 54 25 16 9 29 14 42 20 55 25 17 9 30 15 43 20 56 25 18 10 31 15 44 20 57 26 19 10 32 15 45 21 58 26 20 10 33 16 46 21 59 27 21 11 34 16 47 22 60 27 22 11 35 17 48 22 61 28 23 12 36 17 49 23 62 28 24 12 37 17 50 23 63 28 path from w to s in T . Clearly l(P ) = l(P ) and then noting by V2 the set of vertices of these paths, d(V2 ) and d(V \V2 ) are unchanged. Moreover d(V2 , V \V2 ) is increased by |V \V1 | · l(P3 ) · l(P4 ) > 0.

Hansen, Automated Comparison of Graph Invariants. Les Cahiers du GERAD, G–2005–40, Technical report, HEC Montr´eal. A. E. A. Smith, Using Minimum Degree to Bound Average Distance. Discrete Math. 226(1–3) (2001), 365–371. [4] C. Berge, Graphs and Hypergraphs. North-Holland, Amsterdam, 1976. [5] G. Caporossi, P. Hansen, Variable Neighborhood Search for Extremal Graphs. I. The AutoGraphiX System. Discrete Math. 212(1–2) (2000), 29–44. 36 M. Aouchiche and P. Hansen [6] G. Caporossi, P. Hansen, Variable Neighborhood Search for Extremal Graphs.

Due to the apparent irregularity of optimal values of ∆, no algebraic conjecture could be found. To prove that each value of ∆ given in Table 1 is optimal for the corresponding order n, it is sufficient to show that it maximizes the function fn (∆) = = ∆ · l(∆, n) (n − ∆ + 1)(n − ∆ + 2)(n − 4∆ − 3) + 12(∆ − 1)(∆ − 2) ∆· , 3n(n − 1) 30 M. Aouchiche and P. Hansen where l(∆, n) denotes the average distance in a comet of order n and maximum degree ∆. For example if n = 12, we have −4∆4 + 123∆3 − 989∆2 + 1650∆ .

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0-Dual Closures for Several Classes of Graphs by Ainouche A., Schiermeyer I.


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