Let $G$ be a simple finite connected graph with vertex set $V(G) = \{v_1,v_2,\ldots,v_n\}$. Denote the degree of vertex $v_i$ by $d_i$ for all $1 \leq i \leq n$. The Randi\'c matrix of $G$, denoted by $R(G) = [r_{i,j}]$, is the $n \times n$ matrix whose $(i,j)$-entry $r_{i,j}$ is $r_{i,j} = 1/\sqrt{d_id_j}$ if $v_i$ and $v_j$ are adjacent in $G$ and 0 otherwise. A tree is a connected acyclic graph. A level-wise regular tree is a tree rooted at one vertex $r$ or two (adjacent) vertices $r$ and $r'$ in which all vertices with the minimum distance $i$ from $r$ or $r'$ have the same degree $m_i$ for $0 \leq i \leq h$, where $h$ is the height of $T$. In this paper, we give a complete characterization of the eigenvalues with their multiplicity of the Randi\'c matrix of level-wise regular trees. We prove that the eigenvalues of the Randi\'c matrix of a level-wise regular tree are the eigenvalues of the particular tridiagonal matrices, which are formed using the degree sequence $(m_0,m_1,\ldots,m_{h-1})$ of level-wise regular trees.

翻译：设$G$为具有顶点集$V(G) = \{v_1,v_2,\ldots,v_n\}$的简单有限连通图。记顶点$v_i$的度数为$d_i$（$1 \leq i \leq n$）。$G$的Randić矩阵记为$R(G) = [r_{i,j}]$，是$n \times n$矩阵，其$(i,j)$元$r_{i,j}$定义为：若$v_i$与$v_j$在$G$中相邻，则$r_{i,j} = 1/\sqrt{d_id_j}$，否则为0。树是连通无环图。层级正则树T是一棵以单个顶点$r$或两个（相邻）顶点$r$与$r'$为根，且所有与$r$或$r'$的最小距离为$i$的顶点均具有相同度数$m_i$（$0 \leq i \leq h$）的树，其中$h$为树$T$的高度。本文系统刻画了层级正则树Randić矩阵的特征值及其重数。我们证明，层级正则树Randić矩阵的特征值即为特定三对角矩阵的特征值，该三对角矩阵由层级正则树的度数序列$(m_0,m_1,\ldots,m_{h-1})$构造而成。