The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by G_n,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984–996] conjectured that among all graphs in G_n,m, n ≥ 6 and n − 1 ≤ m ≤ 2(n − 2), the graphs with minimum energy are the star S_n with m−n + 1 additional edges all connected to the same vertices for $m\le n+\lfloor (n-7)/2\rfloor$, and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. In this paper, we provide a new approach to investigate the conjecture above. Especially, we determine the unique tetracyclic graph having minimum energy.

图的能量定义为所有特征值相对于其邻接矩阵的绝对值之和。用G _{n , m}表示具有n 个顶点和m条边的所有连通图的集合。卡波罗西等人。[Caporossi G、Cvetkovi D、Gutman I 等人。极值图的变量邻域搜索。2. 求带外能的图。J Chem Inf Comput Sci。1999;39:984–996] 推测在G _{n , m} , n  ≥ 6 和n  − 1 ≤  m  ≤ 2( n  − 2) 中的所有图中，能量最小的图是星S _n与m − n  + 1 条附加边都连接到相同的顶点$米\le n+\lfloor (n-7)/\mathrm{2个}\rfloor$，以及一侧有两个顶点的二分图，其中一个顶点连接到另一侧的所有顶点，否则。在本文中，我们提供了一种新方法来研究上述猜想。特别是，我们确定了具有最小能量的唯一四环图。