Spectral properties of edge Laplacian matrix

Let $N(X)$ be the Laplacian matrix of a directed graph obtained from the edge adjacency matrix of a graph $X\mathrm{.}$ In this work, we study the bipartiteness property of the graph with the help of $N(X)\mathrm{.}$ We computed the spectrum of the edge Laplacian matrix for the regular graphs, the complete bipartite graphs, and the trees. Further, it is proved that given a graph $X,$ the characteristic polynomial of $N(X)$ divides the characteristic polynomial of $N(X^{\mathrm{\prime }\mathrm{\prime }}),$ where $X^{\mathrm{\prime }\mathrm{\prime }}$ denote the Kronecker double cover of $X\mathrm{.}$