论文标题
在距离拉普拉斯散布和图形的维纳指数上
On distance Laplacian spread and Wiener index of a graph
论文作者
论文摘要
令$ g $为简单连接的简单订单$ n $的图形。距离拉普拉斯矩阵$ d^{l}（g）$定义为$ d^l（g）= diag（tr）-d（g）$，其中$ diag（tr）$是顶点传输的对角矩阵，$ d（g）$是$ g $ $ g $的距离矩阵。 $ d^{l}（g）$的特征值是$ g $的距离laplacian特征值，由$ \ partial_ {1}^{l}^{l}（g），\ partial_ {2}^{2}^{l}^{l}（g），\ dots，\ partial_}连接的图形$ g $的\ textit {远程laplacian spread} $ dls（g）$是最大和第二小的距离laplacian eigenvalues，也就是说，$ \ partial_ {1}^{1}^{l}^{l}（g）（g） - \ partial_ partial_ \ partial_ {n-1}^n-1}^l}^{l} $。我们从Wiener索引$ W（g）$，订单$ n $和最大传输度$ tr_ {max}（g）$ of $ g $的$ dls（g）$获得界限，并表征了极端图。我们获得了$ dls（g）$的两个下限，这是图表的顺序，直径和维也纳索引，而第二个则是图表的顺序，最高程度和独立性数。对于已连接的$ k-partite $ Graph $ g $，$ k \ leq n-1 $，$ n $顶点已断开连接，我们表明 $ dls（g）\ geq \ big \ lfloor \ frac {n} {k} \ big big \ rfloor $具有公平性时，并且只有$ g $是$ g $ as $ complete〜k-partite $图形，每个独立班级具有相同的独立班级和$ n \ equiv equiv 0 \ equiv pmod k $。
Let $G$ be a simple connected simple graph of order $n$. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G), \partial_{2}^{L}(G),\dots,\partial_{n}^{L}(G)$. The \textit{ distance Laplacian spread} $DLS(G)$ of a connected graph $G$ is the difference between largest and second smallest distance Laplacian eigenvalues, that is, $\partial_{1}^{L}(G)-\partial_{n-1}^{L}(G)$. We obtain bounds for $DLS(G)$ in terms of the Wiener index $W(G)$, order $n$ and the maximum transmission degree $Tr_{max}(G)$ of $G$ and characterize the extremal graphs. We obtain two lower bounds for $DLS(G)$, the first one in terms of the order, diameter and the Wiener index of the graph, and the second one in terms of the order, maximum degree and the independence number of the graph. For a connected $ k-partite$ graph $G$, $k\leq n-1$, with $n$ vertices having disconnected complement, we show that $ DLS(G)\geq \Big \lfloor \frac{n}{k}\Big \rfloor$ with equality if and only if $G$ is a $complete ~ k-partite$ graph having cardinality of each independent class same and $n \equiv 0 \pmod k$.
