The variation of the sum of edge lengths in linear arrangements of trees Ramon Ferrer-i-Cancho 1 , Carlos G´ omez-Rodr´ ıguez 2 and Juan Luis Esteban 3 1 Complexity & Quantitative Linguistics Lab, LARCA Research Group Departament de Ci` encies de la Computaci´ o, Universitat Polit` ecnica de Catalunya, Campus Nord, Edifici Omega, Jordi Girona Salgado 1-3. 08034 Barcelona, Catalonia (Spain) 2 Universidade da Coru˜ na, CITIC FASTPARSE Lab, LyS Research Group Departamento de Ciencias de la Computaci´ on y Tecnolog´ ıas de la Informaci´ on, FacultadedeInform´atica, Campus de A Coru˜ na, 15071 A Coru˜ na, Spain 3 Departament de Ci` encies de la Computaci´ o, Universitat Polit` ecnica de Catalunya, Campus Nord, Edifici Omega, Jordi Girona Salgado 1-3. 08034 Barcelona, Catalonia (Spain) E-mail: rferrericancho@cs.upc.edu, cgomezr@udc.es, esteban@cs.upc.edu Abstract. A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems on the sum of edge lengths in trees of a fixed size: the minimum and the maximum value of the sum for specific trees, the minimum and the maximum in classes of trees (bistar trees and caterpillar trees) and finally the minimum and the maximum for any tree. We establish some foundations for research on optimality scores for spatial networks in one dimension. PACS numbers: 89.75.Hc Networks and genealogical trees 89.75.Da Systems obeying scaling laws 89.75.Fb Structures and organization in complex systems Contents 1 Introduction 3 2 Research problems and review 7 2.1 D t min and D t max in specific trees ....................... 7 2.2 D t min and D t max in classes of trees ...................... 9 arXiv:2006.14069v2 [cs.DM] 19 Oct 2020