The Laplacian Minimum Efficient Dominating Energy of a Graph

Abstract

For a graph $G$, a subset $D$ of $V(G)$ is called an efficient dominating set for $G$ if for every vertex $v \in V(G)$, there is exactly one $d \in D$ dominating $v$. The efficient domination number $\gamma_{ED}(G)$ is the minimum cardinality of a efficient dominating set. In this paper we introduce the concept of Laplacian minimum efficient dominating energy $LE_{ED}(G)$ of a graph $G$ and computed Laplacian minimum efficient dominating energies of some standard graphs. Upper and lower bounds for $LE_{ED}(G)$ are established.