2024/01/04 by Punit Vadher, Vadher, Punit, Devsi Bantva +1
Chemistry · Materials Science · Mathematics · #05C05 #05C50 #15A18 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #Phase-change materials and chalcogenides #Synthesis and Properties of Aromatic Compounds
paper · pdf · doi:10.48550/arxiv.2401.02078
openalex publication_date 2024/01/04 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28
Let G be a simple finite connected graph with vertex set V(G) = \v1,v2,…,vn\. Denote the degree of vertex vi by di for all 1 ≤ i ≤ n. The Randić matrix of G, denoted by R(G) = [ri,j], is the n × n matrix whose (i,j)-entry ri,j is ri,j = 1/√(didj) if vi and vj 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 mi for 0 ≤ i ≤ 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 (m0,m1,…,mh-1) of level-wise regular trees.