The minimum vv-coloring Laplacian energy of a graph

Let B(G) denote the set of all blocks of a graph G. Two vertices are vv-adjacent if they incident on the same block. Then vv-degree of a vertex u, d_vv(u) is the number vertices vv-adjacent to the vertex u. In this paper we introduce new kind of graph energy, the minimum vv-coloring Laplacian energy of a graph denoting it as LE_cvv(G). It depends both on underlying graph of G and its particular colors on its vertices of G. We studied some of the properties of LE_cvv(G) and bounds for LE_cvv(G) are established.