Structural properties of hyperideal-based zero-divisor graphs over noncommutative hyperrings

In this paper, we study hyperideal-based zero-divisor graphs ${\mathrm{\Gamma }}_I(R)$ associated with noncommutative hyperrings. We investigate fundamental structural properties of these graphs under the given adjacency condition, including connectivity, diameter bounds, and coloring parameters. For finite hyperrings, we characterize connectedness of ${\mathrm{\Gamma }}_I(R)$ using the rank of the Laplacian matrix. We also establish general bounds on the diameter and identify conditions under which the diameter is at most two. Furthermore, we introduce algebraic measures such as the extended chromatic number and an asymmetry index to capture effects arising from noncommutativity.Several results illustrate how asymmetry in hypermultiplication influences graph structure, including possible differences between classical and extended coloring.