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

Articles in Press

Document Type : Special Issue for ICGTA23

Authors

1 Gujarat Technogical University, Ahmedabad - 382424, Gujarat (INDIA)

2 Lukhdhirji Engineering College, Morvi - 363642, Gujarat (INDIA)

Abstract

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 $\R(G) = [r_{i,j}]$ 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.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 05 September 2024

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