图能量的秩-平均度界
Rank-Average Degree Bound for Graph Energy
浏览论文内容
中文总结 AI 辅助
该研究证明了简单图能量的秩-平均度下界，刻画了极图，并解决了非奇异图能量的五个猜想下界。
中文摘要 AI 辅助
我们证明了，对于任意阶数$n\ge5$的简单图$G$，其能量${\mathcal E}(G)$满足${\mathcal E}\ge r(G)+\bar d(G)-1$，其中$r(G)$和$\bar d(G)$分别表示$G$的邻接矩阵的秩和平均度。我们还刻画了所有极图。作为推论，我们的结果解决了非奇异图能量的五个先前在给定范围内被猜想的下界，即：${\mathcal E}(G)\ge n-1+\bar d(G)$，${\mathcal E}(G)\ge\Delta(G)+\delta(G)$，${\mathcal E}(G)\ge2\sqrt{\bar d(G)(n-1)}$，${\mathcal E}(G)\ge\frac{M_1(G)}{m}$，${\mathcal E}(G)\ge\frac{M_1(G)}{2m}+\frac{2m}{n}$，其中$m$是边数，$\Delta(G)$和$\delta(G)$是最大度和最小度，第一Zagreb指数$M_1(G)$是度平方和。
英文摘要
We prove that the energy ${\mathcal E}(G)$ of any simple graph $G$ of order $n\ge5$ satisfies \[ {\mathcal E}\ge r(G)+\bar d(G)-1, \] where $r(G)$ and $\bar d(G)$ denote, respectively, the rank of the adjacency matrix and the average degree of $G$. We also characterize all extremal graphs. As consequences, our result settles five previously conjectured lower bounds for the energy of nonsingular graphs in their stated ranges, namely \[ \begin{gathered} {\mathcal E}(G)\ge n-1+\bar d(G),\qquad {\mathcal E}(G)\geΔ(G)+δ(G),\qquad {\mathcal E}(G)\ge2\sqrt{\bar d(G) (n-1)},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{m},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{2m}+\frac{2m}{n}, \end{gathered} \] where $m$ is the number of edges, $Δ(G)$ and $δ(G)$ are the maximum and minimum degrees, and the first Zagreb index $ M_1(G)$ is the sum of degree squares.
