The minimum vertex-vertex dominating Laplacian energy of a graph

Let B(G) denote the set of all blocks of a graph G. Two vertices are said to vv-dominate each other if they are vertices of the same block. A set D ⊆ V is said to be vertex-vertex dominating set (vv-dominating set) if every vertex in G is vv-dominated by some vertex in D. The vv-domination number γvv = γvv(G) is the cardinality of the minimum vv-dominating set of G. In this paper, we introduce new kind of graph energy, the minimum vv-dominating Laplacian energy of a graph denoting it as LEvv(G). It depends both on the underlying graph of G and the particular minimum vv-dominating set of G. Upper and lower bounds for LEvv(G) are established and we also obtain the minimum vv-dominating Laplacian energy of some family of graphs.