论文标题
关于图形的相互距离laplacian矩阵的特征值的分布
On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs
论文作者
论文摘要
连接图$ g $的相互距离拉普拉斯矩阵定义为$ rd^l（g）= rt（g）-rd（g）$，其中$ rt（g）$是对角线的对角线矩阵，是互惠距离的距离，$ rd（g）$是Harary Matrix。由于$ rd^l（g）$是一个真正的对称矩阵，我们将其特征值表示为 $λ_1（rd^l（g））\geqλ_2（rd^l（g））\ geq \ dots \geqλ_n（rd^l（g））$。 $ rd^l（g）$的最大特征值$λ_1（rd^l（g））$称为相互距离laplacian光谱半径。在本文中，我们证明$ n $的多重性是$ rd^l（g）$的相互距离laplacian eigenvalue，恰好比补充图$ \ bar {g} $ of $ g $的组件数量少一个。我们表明，完整的两部分图的类别最大化了所有具有$ n $ dertices的两部分图中的相互距离laplacian光谱半径。另外，我们证明了星形$ s_n $是独特的图形，在带有$ n $顶点的树类中具有最大倒数laplacian光谱半径。我们确定了几个知名图的相互距离拉普拉斯频谱。我们证明了完整的图形$ k_n $，$ k_n-e $，star $ s_n $，完整平衡的双分子图$ k _ {\ frac {n} {2} {2}，\ frac {n} {2}}} $和完整的拆分图$ cs（n，α）$是$ cs（n，α）$。
The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RT(G)-RD(G)$, where $RT(G)$ is the diagonal matrix of reciprocal distance degrees and $RD(G)$ is the Harary matrix. Since $RD^L(G)$ is a real symmetric matrix, we denote its eigenvalues as $λ_1(RD^L(G))\geq λ_2(RD^L(G))\geq \dots \geq λ_n(RD^L(G))$. The largest eigenvalue $λ_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of $n$ as a reciprocal distance Laplacian eigenvalue of $RD^L(G)$ is exactly one less than the number of components in the complement graph $\bar{G}$ of $G$. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with $n$ vertices. Also, we show that the star graph $S_n$ is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with $n$ vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph $K_n$, $K_n-e$, the star $S_n$, the complete balanced bipartite graph $K_{\frac{n}{2},\frac{n}{2}}$ and the complete split graph $CS(n,α)$ are all determined from the $RD^L$-spectrum.
