关于图的$A_α$-矩阵的特征值与能量
On the eigenvalues and energy of the $A_α$-matrix of graphs
AI总结：
本文研究了强正则图的$A_α$-矩阵特征值，给出了谱半径的度数边界，证明了完全图是唯一恰有两个不同$A_α$-特征值的图，并导出了$A_α$-能量的上下界及其极值情形。
摘要：
对于图$G$，广义邻接矩阵$A_α(G)$是对角矩阵$D(G)$与邻接矩阵$A(G)$的凸组合，定义为$A_α(G)=αD(G)+(1-α) A(G)$，其中$0\leq α\leq 1$。人们发现该矩阵在融合图$G$的$A(G)$与无符号拉普拉斯矩阵$Q(G)$的谱理论方面非常有用。广义邻接能量或$A_α$-能量是图$G$的$A_α$-特征值的平均偏差，定义为$E(A_α(G))=\sum_{i=1}^{n}|p_i-\frac{2αm}{n}|$，其中$p_i$为$G$的$A_α$-特征值。在本文中，我们研究了强正则图$G$的$A_α$-特征值。我们观察到$A_α$-谱半径$p_1$满足$δ(G)\leq p_1 \leq Δ(G)$，其中$δ(G)$和$Δ(G)$分别是$G$的最小度和最大度。进一步，我们证明了完全图是唯一恰好具有两个不同$A_α$-特征值的图。我们得到了用$G$的阶、大小和极值度表示的$A_α$-能量的上下界，并讨论了这些界的极值情形。
英文摘要：
For a graph $G$, the generalized adjacency matrix $A_α(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_α(G)=αD(G)+(1-α) A(G)$ for $0\leq α\leq 1$. This matrix has been found to be useful in merging the spectral theories of $A(G)$ and the signless Laplacian matrix $Q(G)$ of the graph $G$. The generalized adjacency energy or $A_α$-energy is the mean deviation of the $A_α$-eigenvalues of $G$ and is defined as $E(A_α(G))=\sum_{i=1}^{n}|p_i-\frac{2αm}{n}|$, where $p_i$'s are $A_α$-eigenvalues of $G$. In this paper, we investigate the $A_α$-eigenvalues of a strongly regular graph $G$. We observe that $A_α$-spectral radius $p_1$ satisfies $δ(G)\leq p_1 \leq Δ(G)$, where $δ(G)$ and $Δ(G)$ are, respectively, the smallest and the largest degrees of $G$. Further, we show that the complete graph is the only graph to have exactly two distinct $A_α$-eigenvalues. We obtain lower and upper bounds of $A_α$-energy in terms of order, size and extremal degrees of $G$. We also discuss the extremal cases of these bounds.
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