Abstract
Let \( G \) be a simple graph of order \( n \). The eigenvalue of a graph \( G \) is the eigenvalue of its adjacency matrix. The energy \( \mathcal{E}(G) \) of \( G \) is the sum of absolute values of its eigenvalues. A graph \( G \) of order \( n \) is orderenergetic if \( \mathcal{E}(G)=n \). The algebraic multiplicity of the number zero in the spectrum of \( G \) is referred to as its nullity, and is denoted by \( \eta \). In this paper, we show that if the cycle \( C_{4} \) is not an induced subgraph of a graph \( G \) with nullity \( \eta=3 \), 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 \( k\ge0 \).
