Local energy and vertex deletion for Aα-matrices
Abstract
We study the change in the trace norm and the centred energy of the Aα-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<α<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≤α<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<α<1. Examples show that centred local energy can be negative and that the degree deviation term cannot be omitted in general.
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BibTeX
@misc{kumar2026local,
title = {{Local energy and vertex deletion for $A_α$-matrices}},
author = {Ravinder Kumar and Piyush Sharma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24517},
eprint = {2610.24517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24517},
}
RIS
TY - UNPB
TI - Local energy and vertex deletion for A_α-matrices
AU - Ravinder Kumar
AU - Piyush Sharma
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24517
UR - https://arxiv.org/abs/2610.24517
AB - We study the change in the trace norm and the centred energy of the A_α-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<α<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≤α<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<α<1. Examples show that centred local energy can be negative and that the degree deviation term cannot be omitted in general.
ER -