A Colour-Casimir Adjacency Matrix Approach to Fully-Heavy Tetraquarks

Abstract: We present a phenomenological framework for fully-heavy tetraquark spectroscopy based on the spectral theory of weighted graphs. The four valence partons are vertices of the complete graph $K_4$, with edge weights determined by colour-Casimir factors in the two colour-singlet diquark-antidiquark channels, $\bar{3}\otimes3$ and $6\otimes\bar{6}$. A physical state is described as a coherent mixture of these channels through one mixing angle. Unlike an earlier Laplacian-based version, we obtain the spectrum by directly diagonalising the colour-weighted adjacency matrix. We show analytically that the two constructions are inequivalent and that the Laplacian reverses the physically expected relation between colour attraction and mass ordering. Using $X(6900)$, $X(7100)$, and the tentative $X(7200)$ to determine the mixing angle, energy scale, and effective charm mass, we find $α=0.740$, $γ=23.2$ MeV, and $m_c=1.77$ GeV. Since three parameters are fixed by three inputs, this calibration is not an independent statistical test. The predictive content comes from applying the same parameters elsewhere: the all-bottom ground state is predicted at 18.7-18.9 GeV, while the colour-only model under-binds $T_{cc}<sup>+$ by about 93 MeV. This discrepancy provides a quantitative indication of long-range molecular dynamics beyond a compact four-parton colour graph. We discuss the tentative nature of $X(7200)$, the tension with the CMS radial-excitation interpretation, and the limitations of a static model without spin, orbital, or decay dynamics.