On the Zagreb Domination Energy of a Graph

Abstract

A degree-domination based matrix, termed the Zagreb domination matrix, is introduced for a
simple connected graph G. Its spectral properties dene a new invariant called the Zagreb domination energy,
with bounds derived in terms of order, size, degrees, and domination degrees. Exact formulas and inequalities
are established for standard and k􀀀regular domination{regular graphs. The study further examines the eect
of t􀀀splitting and t􀀀shadow operations, linking eigenvalue inequalities with domination theory.