Characterization of the BJ-Orthograph Radii in a Certain Class of C*-Algebra

In this paper, we investigate the properties of the radius of the BJ-orthograph within the context of finite-dimensional C ∗ -algebras. The BJ-orthograph is a combinatorial structure derived from the orthogonality relations between elements in a C ∗ -algebra, named after its conceptual roots in Birkhoff-James orthogonality. Our primary focus is to establish a comprehensive understanding of the radius of this orthograph, which serves as a measure of the ”distance” from the center to the furthest vertex in this graph-theoretic representation. We begin by defining the BJ-orthograph and outlining its construction in finite-dimensional C ∗ -algebras. Subsequently, we explore the mathematical framework necessary for analyzing its radius, including relevant graph-theoretic and algebraic concepts. Through a series of theorems and lemmas, we derive explicit formulas and bounds for the radius of the BJ-orthograph, leveraging properties unique to C∗ -algebras such as the spectral radius and norm properties. Our results demonstrate how the algebraic structure and dimensionality of the C ∗ -algebra influence the radius of the BJ-orthograph. We provide illustrative examples to highlight these relationships and discuss potential applications in quantum information theory and operator algebras. Finally, we propose several open questions and directions for future research, aiming to extend our findings to broader classes of C ∗ -algebras and other orthogonality-based graphs.