Energy of graph

Energy of graph --- энергия графа.

Let [math]\displaystyle{ G }[/math] be a graph possessing [math]\displaystyle{ n }[/math] vertices and [math]\displaystyle{ m }[/math] edges. Let [math]\displaystyle{ \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} }[/math], be the eigenvalues of the adjacency matrix of [math]\displaystyle{ G }[/math]. The energy of [math]\displaystyle{ G }[/math] is defined as follows

[math]\displaystyle{ {\mathcal E}(G) = \sum_{i=1}^{n} |\lambda_{i}|. }[/math]

The eigenvalues of the [math]\displaystyle{ n }[/math]-vertex complete graph [math]\displaystyle{ K_{n} }[/math] are [math]\displaystyle{ \lambda_{1} = n-1 }[/math], [math]\displaystyle{ \lambda_{2} = \lambda_{3} = \ldots = \lambda_{n} = -1 }[/math]. Therefore, the energy [math]\displaystyle{ {\mathcal E}(K_{n}) = 2n-2 }[/math].

Graphs with the property [math]\displaystyle{ {\mathcal E}(G) \gt 2n - 2 }[/math] are called hyperenergetic graphs; such graphs exist for all [math]\displaystyle{ n \geq 8 }[/math].