If a graph $G$ of order $n$ has the Laplacian energy same as that of complete graph $K_n$ then $G$ is said to be $L$-borderenergeic graph. It is interesting and challenging as well to identify the graphs which are $L$-borderenergetic as only few graphs are known to be $L$-borderenergetic. In the present work we have investigated a sequence of $L$-borderenergetic graphs and also devise a procedure to find sequence of $L$-borderenergetic graphs from the known $L$-borderenergetic graph.