论文标题
独立编号方面的$a_α$索引上的尖锐界限
Sharp bounds on the $A_α$-index of graphs in terms of the independence number
论文作者
论文摘要
鉴于图$ g $，$ g $的邻接矩阵和对角矩阵分别用$ a（g）$和$ d（g）$表示。在2017年，Nikiforov \ cite {0007}提出了$a_α$ -matrix：$a_α（g）=αd（g）+（1-α）a（g），$α\ in [0，1] $。这个小说矩阵的最大特征值称为$ g $的$a_α$索引。在本文中，我们在$ n $ vertex图中具有最低$a_α$ index的图表，而独立数字$ i $ for $α\ in [0,1）$，其中$ i = 1，\ lfloor \ frac {n} {2} \ rfloor，\ lceil \ frac {n} {2} {2} {2} \ rceil，{\ lfloor \ frac {n} $α\ in [0，\ frac {3} {4} {]}。 $a_α$ - 索引，$α\ in [\ frac {1} {2}，1）。$
Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} proposed the $A_α$-matrix: $A_α(G)=αD(G)+(1-α)A(G),$ where $α\in [0, 1]$. The largest eigenvalue of this novel matrix is called the $A_α$-index of $G$. In this paper, we characterize the graphs with minimum $A_α$-index among $n$-vertex graphs with independence number $i$ for $α\in[0,1)$, where $i=1,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil,{\lfloor\frac{n}{2}\rfloor+1},n-3,n-2,n-1,$ whereas for $i=2$ we consider the same problem for $α\in [0,\frac{3}{4}{]}.$ Furthermore, we determine the unique graph (resp. tree) on $n$ vertices with given independence number having the maximum $A_α$-index with $α\in[0,1)$, whereas for the $n$-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum $A_α$-index with $α\in[\frac{1}{2},1).$
