# American Institute of Mathematical Sciences

September  2020, 40(9): 5117-5129. doi: 10.3934/dcds.2020221

## Notes on the values of volume entropy of graphs

 1 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA 2 Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea

* Corresponding author: Seonhee Lim

Received  September 2018 Revised  December 2019 Published  June 2020

Fund Project: This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1601-03 and the National Research Foundation of Korea(NRF) (NRF-2017R1E1A1A03070779, 2017R1A5A1015626)

Volume entropy is an important invariant of metric graphs as well as Riemannian manifolds. In this note, we calculate the change of volume entropy when an edge is added to a metric graph or when a vertex and edges around it are added. In the second part, we estimate the value of the volume entropy which can be used to suggest an algorithm for calculating the persistent volume entropy of graphs.

Citation: Wooyeon Kim, Seonhee Lim. Notes on the values of volume entropy of graphs. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5117-5129. doi: 10.3934/dcds.2020221
##### References:

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##### References:
 [1] John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16. [2] Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075 [3] François Ledrappier, Seonhee Lim. Volume entropy of hyperbolic buildings. Journal of Modern Dynamics, 2010, 4 (1) : 139-165. doi: 10.3934/jmd.2010.4.139 [4] Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325 [5] Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75 [6] Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205 [7] Rui Wang, Rundong Zhao, Emily Ribando-Gros, Jiahui Chen, Yiying Tong, Guo-Wei Wei. HERMES: Persistent spectral graph software. Foundations of Data Science, 2021, 3 (1) : 67-97. doi: 10.3934/fods.2021006 [8] Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789 [9] Gunhild A. Reigstad. Numerical network models and entropy principles for isothermal junction flow. Networks & Heterogeneous Media, 2014, 9 (1) : 65-95. doi: 10.3934/nhm.2014.9.65 [10] Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017 [11] Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237 [12] Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018 [13] José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301 [14] Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226 [15] François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275 [16] Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395 [17] Nicolas Bedaride. Entropy of polyhedral billiard. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89 [18] Karl Petersen, Ibrahim Salama. Entropy on regular trees. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4453-4477. doi: 10.3934/dcds.2020186 [19] Vladimír Špitalský. Local correlation entropy. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249 [20] Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204

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