论文信息 - The average size of a connected vertex set of a k-connected graph

The average size of a connected vertex set of a k-connected graph

The topic is the average order A(G) of a connected induced subgraph of a graph G. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1984, Jamison proved that the average order, over all trees of order n, is minimized by the path Pn, the average being A(Pn) = (n + 2)/3. In 2018, Kroeker, Mol, and Oellermann conjectured that Pn minimizes the average order over all connected graphs G a conjecture that was recently proved. In this short note we show that this lower bound can be improved if the connectivity of G is known. If G is k-connected, then A(G) ≥ n 2 (

Andrew Vince

[1] Ortrud R. Oellermann,et al. On the mean connected induced subgraph order of cographs , 2018, Australas. J Comb..

[2] John Haslegrave,et al. Extremal results on average subtree density of series-reduced trees , 2013, J. Comb. Theory, Ser. B.

[3] Robert E. Jamison,et al. Monotonicity of the mean order of subtrees , 1984, J. Comb. Theory, Ser. B.

[4] Hua Wang,et al. The average order of a subtree of a tree , 2010, J. Comb. Theory, Ser. B.

[5] Yeong-Nan Yeh,et al. Enumeration of subtrees of trees , 2006, Theor. Comput. Sci..

[6] Robert E. Jamison,et al. On the average number of nodes in a subtree of a tree , 1983, J. Comb. Theory, Ser. B.

[7] ANDREW VINCE. The average size of a connected vertex set of a graph - Explicit formulas and open problems , 2021, J. Graph Theory.

[8] Ortrud R. Oellermann,et al. The mean order of sub‐k‐trees of k‐trees , 2018, J. Graph Theory.

[9] Ortrud R. Oellermann,et al. Maximizing the mean subtree order , 2019, J. Graph Theory.

[10] Andrew Vince. A lower bound on the average size of a connected vertex set of a graph , 2022, J. Comb. Theory, Ser. B.

[11] Hua Wang,et al. On the Local and Global Means of Subtree Orders , 2016, J. Graph Theory.