Truncated Clique Reconstruction in Non-Commuting Graphs of Finite Groups with Abelian Centralizers

Abstract

Purpose: This study investigates how much truncated clique information is sufficient to reconstruct the non-commuting graph of a finite group with abelian centralizers. Methodology: The complete multipartite structure of these graphs is encoded by a centralizer partition profile. Clique counts are converted into power sums through Newton identities, and the multiplicities of prescribed part sizes are recovered from a Vandermonde system. The framework is then combined with arithmetic realizability conditions and exterior-square geometry for an exceptional class-three AC-3-group branch. Findings: If the possible part sizes assume d distinct prescribed values, their multiplicities are uniquely determined by the first d power sums, equivalently by a short initial segment of clique data. In the exceptional branch with [P:Z(P)]=3⁸, [C:Z(P)]=3⁴ and C=P′Z(P), the possible non-distinguished centralizers have relative orders 3 and 9. Under the nondegeneracy condition that the associated projective kernel line is not contained in the Klein quadric, the number m₂ of relative-order-9 centralizers satisfies the sharp bound m₂≤162. An explicit AC-3-group of order 3¹² attains equality, with non-commuting graph Γ_P ≅ K_{6480,(162)^2592,(648)^162} and ω(Γ_P)=χ(Γ_P)=2755. Significance: The results connect partial graph reconstruction, moment methods, and group-theoretic realizability, and show that truncated clique data can recover substantial algebraic-combinatorial structure.