Local energy and vertex deletion for $A_\alpha$-matrices

Abstract

We study the change in the trace norm and the centred energy of the $A_\alpha$-matrix of a graph under vertex deletion. A pinching argument proves that the trace norm is nondecreasing under passage from an induced subgraph to the original graph, and gives a lower bound in terms of the edges induced by the deleted vertices. Exact perturbation spectra yield sharp upper bounds for deleting a vertex or an independent set of vertices with a common neighbourhood. For $0<\alpha<1/2$, equality in these upper bounds is characterized by star and complete bipartite components, respectively. For the centred energy, we derive explicit deletion bounds and classify all connected equality graphs for $0\le\alpha<1$: the extremal graphs are stars and complete bipartite graphs for single-vertex and independent-twin deletion, respectively. We also characterize equality in the general-set trace-norm lower bound and prove strictness of the total centred-energy inequality for connected nonregular graphs when $0<\alpha<1$. Examples show that centred local energy can be negative and that the degree deviation term cannot be omitted in general.