## 7.3 Hyperenergetic, Hypoenergetic, and Equienergetic Graphs

### 7.3.1 **Hyperenergetic Graphs**

The energy of the *n*-vertex complete graph *K*_{n} is equal to 2(*n*–1). We call an *n*-vertex graph *G hyperenergetic* if *E*(*G*) > 2(*n*–1). From Nikiforov's Theorem 7.24 we see that almost all graphs are hyperenergetic. Therefore, any search for hyperenergetic graphs appears nowadays a futile task. Yet, before Theorem 7.24 was discovered, a number of such results were obtained. We outline here some of them.

In [7] it was conjectured that the complete graph *K*_{n} had greatest energy among all *n*-vertex graphs. This conjecture was soon shown to be false [44].

The first systematic construction of hyperenergetic graphs was proposed by Walikar, Ramane, and Hampiholi [45], who showed that the line graphs of *K*_{n}, *n* ≥ 5, and of *K*_{n/2, n/2}, *n* ≥ 8, are hyperenergetic. These results were eventually extended to other graphs with a large number of edges [46, 47].

Hou and Gutman [48] showed that the line graph of any (*n*, *m*)-graph, *n* ≥ 5, *m* ≥ 2*n*, is hyperenergetic. Also, the line graph of any bipartite (*n*, *m*)-graph, *n* ≥ 7, *m* ≥ 2(*n* – 1), is hyperenergetic. Some classes of circulant graphs [49–51] as well as Kneser graphs and their complements [52] are hyperenergetic. In fact, almost all circulant graphs are hyperenergetic [49].

Graphs on *n* vertices with fewer than 2*n* – 1 edges are not hyperenergetic [53, 54]. This, in particular, implies that Hückel graphs (graphs representing conjugated molecules [2–4], in which the vertex degrees ...