On the construction of LL-equienergetic graphs

For a graph GG with nn vertices and mm edges, and having Laplacian spectrum μ1,μ2,…,μnμ1,μ2,…,μn and signless Laplacian spectrum μ1+,μ2+,…,μn+, the Laplacian energy and signless Laplacian energy of GG are respectively, defined as LE(G)=∑i=1n|μi−2mn| and LE+(G)=∑i=1n|μi+−2mn|. Two graphs G1G1 and G2G2 of same order are said to be LL-equienergetic if LE(G1)=LE(G2)LE(G1)=LE(G2) and QQ-equienergetic if LE+(G1)=LE+(G2)LE+(G1)=LE+(G2). The problem of constructing graphs having same Laplacian energy was considered by Stevanovic for threshold graphs and by Liu and Liu for those graphs whose order is n≡0n≡0 (mod 7). We consider the problem of constructing LL-equienergetic graphs from any pair of given graphs and we construct sequences of non-cospectral (Laplacian, signless Laplacian) LL-equienergetic and QQ-equienergetic graphs from any pair of graphs having same number of vertices and edges.