Energy of binary labeled graphs

Abstract

‎‎‎Let $G$ be a graph with vertex set $V(G)$ and edge set $X(G)$ and consider the set $A={0,1}$‎. ‎A mapping $l:V(G)longrightarrow A$ is called binary vertex labeling of $G$ and $l(v)$ is called the label of the vertex $v$ under $l$‎. ‎In this paper we introduce a new kind of graph energy for the binary labeled graph‎, ‎the labeled graph energy $E_{l}(G)$‎. ‎It depends on the underlying graph $G$ and on its binary labeling‎, ‎upper and lower bounds for $E_{l}(G)$ are established‎. ‎The labeled energies of a number of well known and much studied families of graphs are computed‎.