论文标题
图形和色度和独立性的距离拉普拉斯特征值
Distance Laplacian eigenvalues of graphs and chromatic and independence number
论文作者
论文摘要
对于连接的图形$ g $ $ n $，令$ diag（tr）$是顶点传输的对角线矩阵，$ d（g）$是$ g $的距离矩阵。 $ g $的距离laplacian矩阵定义为$ d^l（g）= diag（tr）-d（g）$，$ d^{l}（g）$的特征值称为$ g $的距离laplacian特征值。令$ \ partial_ {1}^{l}（g）\ geq \ partial_ {2}^{l}（g）（g）\ geq \ dots \ geq \ geq \ geq \ partial_ {n}^{l}^{l}^{l}（g）$ be laplacian eigenvalues of $ g $。给定一个间隔$ i $，让$ m_ {d^{l}（g）} i $（或简单$ m_ {d^{l}} i $）是$ g $的距离laplacian eigenvalues的数量，这些$ g $位于间隔$ i $中。对于规定的间隔$ i $，我们确定$ m_ {d^{l}} i $在独立数字$α（g）$，色数$χ$，吊坠顶点和直径$ d $的$ g $。特别是，我们证明$ m_ {d^{l}（g）} [n，n+2）\ leqχ-1 $，〜$ m_ {d^{l}（g）} [n，n，n+α（g））\ \ leq n-α（g），我们表明了不平等现象。我们还表明，$ m_ {d^{l}（g）} \ bigG（n，n，n+\ left \ lceil \ frac {n}χ\ right \ rceil \ rceil \ rceil \ bigG）\ leq n- \ leq n- \ lest \ left \ left \ left \ lceil \ laceil \ frac { $ c _ {\ overline {g}} $是$ \ overline {g} $中的组件数，并讨论某些最可能的界限。此外，我们证明$ m_ {d^{l}（g）} [n，n+p）\ leq n-p $，其中$ p \ geq 1 $是吊坠顶点的数量。另外，我们表征了直径$ d \ leq 2 $满足$ m_ {d^{l}（g）}（2n-1,2n）=α（g）-1 = \ frac {n} {2} -1 $的图表。最后，我们提出了一些感兴趣的问题。
For a connected graph $G$ of order $n$, let $Diag(Tr)$ be the diagonal matrix of vertex transmissions and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $D^L(G)=Diag(Tr)-D(G)$ and the eigenvalues of $D^{L}(G)$ are called the distance Laplacian eigenvalues of $G$. Let $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$ be the distance Laplacian eigenvalues of $G$. Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L} } I$) be the number of distance Laplacian eigenvalues of $G$ which lie in the interval $I$. For a prescribed interval $I$, we determine $m_{D^{L} }I$ in terms of independence number $α(G)$, chromatic number $χ$, number of pendant vertices and diameter $d$ of the graph $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq χ-1$, ~$m_{D^{L}(G) }[n,n+α(G))\leq n-α(G)$ and we show that the inequalities are sharp. We also show that $m_{D^{L} (G )}\bigg( n,n+\left\lceil\frac{n}χ\right\rceil\bigg)\leq n- \left\lceil\frac{n}χ\right\rceil-C_{\overline{G}}+1 $, where $C_{\overline{G}}$ is the number of components in $\overline{G}$, and discuss some cases where the bound is best possible. In addition, we prove that $m_{D^{L} (G )}[n,n+p)\leq n-p$, where $p\geq 1$ is the number of pendant vertices. Also, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= α(G)-1=\frac{n}{2}-1$. At the end, we propose some problems of interest.
