A complete characterization of spectra of the Randic matrix of level-wise regular trees

Abstract

Let $G$ be a simple finite connected graph with vertex set $V(G)=\{v_1,v_2,\dots ,v_n\}$. Denote the degree of vertex $v_i$ by $d_i$ for all $1\le i\le n$. The Randić matrix of $G$, denoted by $R(G)=[r_{i,j}]$, is the $n\times n$ matrix whose $(i,j)$-entry $r_{i,j}$ is $r_{i,j}=1/\sqrt{d_i{d}_j}$ 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 $r^{\prime }$ in which all vertices with the minimum distance $i$ from $r$ or $r^{\prime }$ have the same degree $m_i$ for $0\le i\le h$, 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 $(m_0,m_1,\dots ,m_{h-1})$ of level-wise regular trees.

20 pages, 2 figures