On Distance Laplacian Energy of Unicyclic and Bicyclic Graphs

For a connected graph G, let DL(G) be its distance Laplacian matrix and λ1(G)≥λ2(G)≥…≥λn−1(G)>λn(G)=0 be its DL eigenvalues. The DL energy of G is defined as DLEG=∑i=1nλi(G)−2WGn, where W(G) is the Wiener index of G. An important problem in graph energy studies is to determine exact formulations of the energy for specific graph classes and their complements. This paper gives the precise DL energy formulations of a class of bicyclic graphs C2(p,q), a class of unicyclic graphs C1(p,q), and their complements. Moreover, we order the graphs C2(p,q) on the basic of λ1, λn−1, and consider the same problems for their complements. And the ordering of the graphs C1(p,q) on the basic of λn−1 and the ordering of their complements on the basics of λ1, λn−1 and the DL energy are obtained.