Mean Distance on Metric Graphs

被引:0
|
作者
Baptista, Luis N. [1 ,2 ]
Kennedy, James B. [1 ,2 ]
Mugnolo, Delio [3 ]
机构
[1] Univ Lisbon, Fac Ciencias, Dept Matemat, Edificio C6, P-1749016 Lisbon, Portugal
[2] Inst Super Tecn, Grp Fis Matemat, Ave Rovisco Pais, P-1049001 Lisbon, Portugal
[3] Fernuniv, Fak Math & Informat, Lehrgebiet Anal, D-58084 Hagen, Germany
关键词
Metric graph; Mean distance; Graph surgery; Laplacian; Spectral gap; Coarea formula; EIGENVALUES; DIAMETER;
D O I
10.1007/s12220-024-01574-0
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of the spectral gap of the graph Laplacian with standard vertex conditions: it is maximised among all graphs of fixed length by the path graph (interval), or by the loop in the restricted class of doubly connected graphs, and it is minimised among all graphs of fixed length and number of edges by the equilateral flower graph. We also establish bounds for the correctly scaled product of the spectral gap and the square of the mean distance which depend only on combinatorial, and not metric, features of the graph. This raises the open question whether this product admits absolute upper and lower bounds valid on all compact metric graphs.
引用
收藏
页数:25
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