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 ⁢ d e t ⁡ ( 𝜆 2 ⁢ 𝐼 𝑛 − 𝜆 ⁢ 𝐵 𝜑 − 𝑅 𝜑 ) . Here 𝑛 = | 𝑉 ⁡ ( 𝐺 ) | , 𝑀 = ∑ 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 2 ⁢ 𝑛 -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.