On Minimum Dominating Minimum Degree Energy of Graphs

Let [Formula: see text] be a simple graph with [Formula: see text], minimum degree [Formula: see text], and domination number [Formula: see text]. The Minimum Dominating Minimum Degree Matrix, denoted by [Formula: see text], is introduced as a domination–constrained refinement of the classical minimum degree matrix, where the diagonal entries are defined with respect to a fixed minimum dominating set [Formula: see text] satisfying [Formula: see text]. If [Formula: see text] are the eigenvalues of [Formula: see text], then the corresponding Minimum Dominating Minimum Degree Energy is defined as [Formula: see text] Fundamental algebraic and spectral properties of [Formula: see text] are established, including trace identities and a general trace–square formula that yields lower bounds for [Formula: see text]. The spectral behaviour under generalized graph composition is analyzed using Kronecker product techniques, leading to a decomposition theorem and an asymptotic energy formula for cyclic constructions. Explicit spectral characterizations and energy bounds are established for [Formula: see text]–regular graphs, with particular emphasis on the cubic ([Formula: see text]) and quartic ([Formula: see text]) cases. These results extend degree–based spectral graph theory by systematically incorporating domination parameters into matrix–based energy invariants.