The α\alpha-index of graphs without intersecting triangles/quadrangles as a minor

The $A_{\alpha}$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)$, where $0\leq\alpha \leq1$. The $\alpha$-index of $G$ is the largest eigenvalue of $A_\alpha(G)$. Particularly, the matrix $A_0(G)$ (resp. $2A_{\frac{1}{2}}(G)$) is exactly the adjacency matrix (resp. signless Laplacian matrix) of $G$. He, Li and Feng [arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency spectral radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively. Motivated by the above results of He, Li and Feng, in this paper we characterize the extremal graphs with maximum $\alpha$-index among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor for any $0<\alpha<1$, respectively. As by-products, we determine the extremal graphs with maximum signless Laplacian radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively.Comment: 15 page

Dissipative Effects on Quantum Sticking

Using variational mean-field theory, many-body dissipative effects on the threshold law for quantum sticking and reflection of neutral and charged particles are examined. For the case of an ohmic bosonic bath, we study the effects of the infrared divergence on the probability of sticking and obtain a non-perturbative expression for the sticking rate. We find that for weak dissipative coupling $\alpha$, the low energy threshold laws for quantum sticking are modified by an infrared singularity in the bath. The sticking probability for a neutral particle with incident energy $E\to 0$ behaves asymptotically as ${\it s}\sim E^{(1+\alpha)/2(1-\alpha)}$; for a charged particle, we obtain ${\it s}\sim E^{\alpha/2(1-\alpha)}$. Thus, "quantum mirrors" --surfaces that become perfectly reflective to particles with incident energies asymptotically approaching zero-- can also exist for charged particles.Comment: 10 pages, 0 fig