The minimum vertex–vertex dominating Laplacian energy of a graph

Abstract

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