Journal Pre-proof Physical assessments on the invariant region in multi-route reaction mechanism Mehboob Ali, Muhammad Shahzad, Faisal Sultan, Waqar Azeem Khan PII: S0378-4371(19)31433-5 DOI: https://doi.org/10.1016/j.physa.2019.122499 Reference: PHYSA 122499 To appear in: Physica A Received date : 6 April 2019 Revised date : 29 April 2019 Please cite this article as: M. Ali, M. Shahzad, F. Sultan et al., Physical assessments on the invariant region in multi-route reaction mechanism, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.122499. This is a PDF ﬁle of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the deﬁnitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its ﬁnal form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.