Interplay Between Node and Edge Laplacians Through Spectral Analysis of Redefined ‘Degree and Adjacency’ Matrices of Digraphs

This work seeks to interconnect the well-known concept of the graph Laplacian with its edge counterpart, the Edge Laplacian with respect to a new and updated definition of edge-degree and edge-adjacency matrix. It focuses on establishing spectral connections not only between these two Laplacians but also between their components, namely the degree matrices and the adjacency matrices. To achieve this interconnection, the edge degree and adjacency matrix have been redefined to not only offer a graph-theoretic interpretation of edge-based analysis but also uniquely link the corresponding edge and adjacency matrices through eigenvalue-based analysis.