论文标题
图形的广义相互距离矩阵
The generalized reciprocal distance matrix of graphs
论文作者
论文摘要
让$ g $是一个简单的无向连接图，使用Harary Matrix $ RD（G）$，它也称为$ G $的相互距离矩阵。 $ g $的倒数距离laplacian矩阵是$ rq（g）= rt（g）+rd（g）$，其中$ rt（g）$表示$ g $的顶点互联母对角线矩阵。本文打算引入一个新的矩阵$rd_α（g）=αrt（g）+（1-α）rd（g）$，$α\ in [0,1] $，以跟踪从$ rd（g）$到$ rq（g）$的逐渐变化。首先，我们完全描述了一些特殊图形的$rd_α（g）$的特征值。然后，我们获得了$RD_α（g）$的服务基本属性，包括涉及相互距离矩阵的光谱半径的不等式，倒数距离laplacian矩阵和$ g $的$rd_α$ -matrix。我们还提供了$RD_α$ -MATRIX的光谱半径的一些下限和上限。最后，我们在所有连接的固定顺序和精确的顶点连接，边缘连接，色数和独立数中分别描绘了$rd_α$ -matrix的最大光谱半径的极端图。
Let $G$ be a simple undirected connected graph with the Harary matrix $RD(G)$, which is also called the reciprocal distance matrix of $G$. The reciprocal distance signless Laplacian matrix of $G$ is $RQ(G)=RT(G)+RD(G)$, where $RT(G)$ denotes the diagonal matrix of the vertex reciprocal transmissions of graph $G$. This paper intends to introduce a new matrix $RD_α(G)=αRT(G)+(1-α)RD(G)$, $α\in [0,1]$, to track the gradual change from $RD(G)$ to $RQ(G)$. First, we describe completely the eigenvalues of $RD_α(G)$ of some special graphs. Then we obtain serval basic properties of $RD_α(G)$ including inequalities that involve the spectral radii of the reciprocal distance matrix, reciprocal distance signless Laplacian matrix and $RD_α$-matrix of $G$. We also provide some lower and upper bounds of the spectral radius of $RD_α$-matrix. Finally, we depict the extremal graphs with maximal spectral radius of the $RD_α$-matrix among all connected graphs of fixed order and precise vertex connectivity, edge connectivity, chromatic number and independence number, respectively.
