Degree-Based Weighted Adjacency Spectra of Thorny Graphs with Positive Multiplicities

Let G be a finite simple graph, let \( m = (m_v)_{v∈V(G)} \) with m_v ≥ 1, and form G^m by attaching m_v pendant vertices at each v. For the weighted adjacency matrix A_φ(G^m) induced by a real symmetric degree-dependent edge weight φ, a generalized Fiedler-type decomposition gives M − n forced zero eigenvalues and \( χ_{A_φ(G^m)}(λ) = λ^{M−n} \) det\( (λ²I_n − λB_φ − R_φ) \), where\( n = |V(G)| \), \( M = Σ_v m_v \), B_φ is the core block formed from the core-vertex degrees in the thorny graph, and R_φ is diagonal with squared pendant-fiber couplings. The decomposition gives an arbitrary-core nullity bound, exact nullity and inertia for nonzero pendant couplings, Cauchy interlacing with B_φ, and a 2n-dimensional compression independent of M. For fixed r ≥ 1, m, and φ, the nonzero weighted spectrum of a uniform attachment to an r-regular core determines the core adjacency spectrum whenever the induced core-edge weight is nonzero. On connected walk-regular cores, let exceptional vertices receive m + q rather than m pendant vertices. For independent exceptional sets S,T with |S| = |T| ≥ 2, all moments below twice the smaller minimum separation agree for every real symmetric φ. If the separations differ, then at the first possible mixed order the moment of the smaller-separation placement minus that of the larger-separation placement is an explicit nonnegative multiple of a sum of squared geodesic counts. If the ordinary–ordinary core-edge weight is nonzero and its square differs from that of the exceptional–ordinary core-edge weight, the spectrum determines the minimum separation. On distance-regular cores, it also determines the closest-pair count. Both the ordinary Sombor and Randić weights satisfy this condition.