Cut Decomposition and Induced-Edge Statistics for Degree-Based Edge Indices Under Thorn Reassignments on Regular Cores

Let \(G\) be an \(r\)-regular graph of order n, r ≥ 1, and form \(G^{(\mathbf{m})}\) by attaching \(m_v\) pendants to each core vertex \(v\). For a symmetric degree-based weight \(\varphi\), define \[ I_\varphi(H) = \sum_{uv \in E(H)} \varphi\bigl(d_H(u), d_H(v)\bigr). \] Fix the thorn multiset \(\{m_v : v \in V(G)\}\), but allow its values to be reassigned among core vertices. A multilevel cut decomposition isolates placement dependence. The cut between the core classes of degrees \(a\) and \(b\) has coefficient \[ -\tfrac12 \, \Delta\varphi(a,b), \quad \text{where } \Delta\varphi(a,b) = \varphi(a,a) - 2\varphi(a,b) + \varphi(b,b). \] For a fixed core and a finite nonempty set \(A\) of attainable degrees, every thorn multiset whose core-vertex degrees lie in \(A\) is reassignment-invariant if and only if \(G\) is complete or the restricted weight matrix is a symmetric sum matrix. If invariance is required for every connected \(r\)-regular core, where r ≥ 2, only the sum-matrix case remains. With two thorn levels, \(I_\varphi\) is affine in the induced-edge count \(e_G(S)\) of the \(k\) vertices receiving the larger number. This relation determines the value set and exact range and gives a criterion for fixed-cardinality invariance. When the coefficient is nonzero, the extremal-placement problem is NP-hard on cubic cores. For fixed \(k\), the mean and variance of the induced-edge count depend on the core only through its order and regular degree. For \(3 \le k \le n-3\), the third factorial moment determines the triangle count, and the index distribution does so as well whenever the affine coefficient is nonzero. For cycles and equal-part complete multipartite cores, the ranges have closed forms. The ordinary, reduced, and Euler Sombor indices are maximized when the heavy-vertex set induces as few core edges as possible, whereas the elliptic Sombor index is maximized when it induces as many as possible. The forgotten index is placement-invariant.