SPECTRAL PROPERTIES OF THE SKEW MAXIMUM DEGREE LAPLACIAN MATRIX OF DIGRAPHS

The concept of graph energy has attracted considerable attention due to its applications in spectral graph theory, combinatorics, chemistry, communication networks, and data science. In recent years, several variants of graph energy have been introduced by employing different graph matrices, including the adjacency matrix, Laplacian matrix, signless Laplacian matrix, and skew-adjacency matrix. Motivated by the skew Laplacian energy of digraphs, we introduce a new degree-based matrix called the skew maximum degree matrix. Using this matrix, we define the skew maximum degree Laplacian matrix of a digraph and study its associated energy. Let \(D(G)\) denote the diagonal matrix of vertex degrees and \(S(G)\) denote the skew maximum degree matrix. Then the skew maximum degree Laplacian matrix is defined by \[ L_{S}(G)=D(G)-S(G). \] The corresponding skew maximum degree Laplacian energy is introduced and investigated. Explicit expressions, spectral properties, lower and upper bounds, extremal results, and connections with classical graph invariants are established. The skew maximum degree Laplacian matrix appears to provide a promising bridge between classical skew spectral graph theory and modern degree-based approaches in mathematics.