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 $X.$ In this work, we study the bipartiteness property of the graph with the help of $N(X).$ 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 $N(X^{\prime\prime}),$ where $X^{\prime\prime}$ denote the Kronecker double cover of $X.$