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arXiv 2005.01037math.COmath.SP
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关于图的$\alpha$-邻接能量与Zagreb指数

On $α$-adjacency energy of graphs and Zagreb index

University of Kashmir（克什米尔大学）

S. Pirzada, Bilal A. Rather, Hilal A. Ganie, Rezwan ul Shaban

备注：
17 pages
发表信息：
AKCE International Journal of Graphs and Combinatorics, Volume 18, Issue 1, 2021, Pages 39-46
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AI总结：

该研究围绕图的$\alpha$-邻接能量展开，推导了其用图的阶数、边数及Zagreb指数表示的上下界，并刻画了达到这些界的极图，丰富了谱图理论相关结论。

摘要：

设$A(G)$为简单连通图$G$的邻接矩阵，$D(G)$为其顶点度对角矩阵。Nikiforov将$D(G)$与$A(G)$的凸组合矩阵$A_α(G)$定义为$A_α(G)=αD(G)+(1-α)A(G)$，其中$0\leq α\leq 1$。若$ρ_{1}\geq ρ_{2}\geq \dots \geq ρ_{n}$是$A_α(G)$的特征值（我们称之为$G$的$\alpha$-邻接特征值），则$G$的$\alpha$-邻接能量定义为$E^{A_α}(G)=\sum_{i=1}^{n}\left|ρ_i-\frac{2αm}{n}\right|$，其中$n$为$G$的阶数，$m$为其边数。我们得到了用$G$的阶数$n$、边数$m$以及与其结构相关的Zagreb指数$Zg(G)$表示的$E^{A_α}(G)$的上下界，并进一步刻画了达到这些界的极图。

英文摘要：

Let $A(G)$ be the adjacency matrix and $D(G)$ be the diagonal matrix of the vertex degrees of a simple connected graph $G$. Nikiforov defined the matrix $A_α(G)$ of the convex combinations of $D(G)$ and $A(G)$ as $A_α(G)=αD(G)+(1-α)A(G)$, for $0\leq α\leq 1$. If $ ρ_{1}\geq ρ_{2}\geq \dots \geq ρ_{n}$ are the eigenvalues of $A_α(G)$ (which we call $α$-adjacency eigenvalues of $G$), the $ α$-adjacency energy of $G$ is defined as $E^{A_α}(G)=\sum_{i=1}^{n}\left|ρ_i-\frac{2αm}{n}\right|$, where $n$ is the order and $m$ is the size of $G$. We obtain the upper and lower bounds for $E^{A_α}(G) $ in terms of order $n$, size $m$ and Zagreb index $Zg(G)$ associated to the structure of $G$. Further, we characterize the extremal graphs attaining these bounds.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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