Book ChapterControl-based Conditions for Graph DistinguishabilityRaza, Muhammad; Ahmad, Obaid Ullah; Shabbir, Mudassir; Koutsoukos, Xenofon; Abbas, WaseemThe graph distinguishability problem investigates whether a graph can be uniquely identified by the spectrum of its adjacency matrix, specifically determining if two graphs with the same spectrum are isomorphic. This issue is central to spectral graph theory and has significant implications for graph machine learning. In this paper, we explore the intricate connections between graph distinguishability and graph controllability–an essential concept in the control of networked systems. Focusing on oriented graphs and their skew-adjacency matrices, we establish controllability-based conditions that ensure their distinguishability. Notably, our conditions are less restrictive than existing methods, enabling a broader class of graphs to satisfy the distinguishability criteria. We illustrate the effectiveness of our results with several examples. Our findings highlight the applications of network control methods in tackling this crucial problem in algebraic graph theory, with implications for machine learning and network design.Society for Industrial and Applied Mathematics2025-01-011062655431 to 38978-1-61197-874-2https://doi.org/10.1137/1.9781611978742.52325416; 2325417National Science Foundation