Abstract
In this chapter we explore recent development on various problems related to graph indices in trees. We focus on indices based on distances between vertices, vertex degrees, or on counting vertex or edge subsets of different kinds. Some of the indices arise naturally in applications, e.g., in chemistry, statistical physics, bioinformatics, and other fields, and connections are also made to other branches of graph theory, such as spectral graph theory. We will be particularly interested in the extremal values (maxima and minima) for different families of trees and the corresponding extremal trees. Moreover, we review results for random trees, consider localized versions of different graph indices and the associated notions of centrality, and finally discuss inverse problems, where one wants to find trees for which a specific graph index has a prescribed value.
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Acknowledgements
This research was supported in part by the NSF DMS contract 1300547 and by the DARPA and AFOSR under the contract FA9550-12-1-0405 (László Székely), National Research Foundation of South Africa, grant number 96236 (Stephan Wagner), and Simons Foundation grant 245307 (Hua Wang).
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Székely, L., Wagner, S., Wang, H. (2016). Problems related to graph indices in trees. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_1
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