On Orderenergetic Graphs

Let G be a simple graph of order n. The eigenvalue of a graph G is the eigenvalue of its adjacency matrix. The energy E(G) of G is the sum of absolute values of its eigenvalues. A graph G of order n is orderenergetic if E(G) = n. The algebraic multiplicity of the number zero in the spectrum of G is referred to as its nullity, and is denoted by η. In this paper, we show that if the cycle C4 is not an induced subgraph of a graph G with nullity η = 3, then G is not orderenergetic. We also obtain some results connecting orderenergetic graphs and minimum degree. Finally, we show that there is a connected orderenergetic graph on 10k + 8 vertices for all k ≥ 0.