a complete characterization of spectra of the randić matrix of level-wise regular trees

Let $g$ be a simple finite connected graph with vertex set $v(g) = {v_1,v_2,ldots,v_n}$ and $d_i$ be the degree of the vertex $v_i$. the randić matrix ${bf r}(g) = [r_{i,j}]$ of graph $g$ is an $n times n$ 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 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 randić matrix of level-wise regular trees. we prove that the eigenvalues of the randić 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.