Color laplacian energy of generalised complements of a graph

The color energy of a graph is defined as sum of absolute color eigenvalues of graph, denoted by E_c(G). Let G_c = (V, E) be a color graph and P = {V₁, V₂, …, V_k } be a partition of V of order k ≥ 1. The k-color complement {G_c}^Pk of G_c is defined as follows: For all V_i and V_j in P, i ≠ j, remove the edges between V_i and V_j and add the edges which are not in G_c such that end vertices have different colors. For each set V_r in the partition P, remove the edges of G_c inside V_r, and add the edges of G_c (the complement of G_c) joining the vertices of V_r. The graph {G_c}^P_k(i) thus obtained is called the k(i)− color complement of G_c with respect to the partition P of V. In this paper, we compute color Laplacian energy of generalised complements of few standard graphs. Color Laplacian energy depends on assignment of colors to the vertices and the partition of V (G).