ABSTRACT
The Randić matrix (short for R-matrix) of a graph G is a symmetric matrix whose
-entry is equal to
if
, and 0 otherwise. The R-eigenvalues of a graph G are the eigenvalues of its Randić matrix R. In this paper, we introduce the star complements of R-matrix of a graph G, especially of a tree T, from which we investigate the R-eigenvalues of T. First, if a tree T has a R-eigenvalue ρ with multiplicity k, then
whenever
. Second, a tree T has at least k+1 pendant edges form an induced matching if it has
as a R-eigenvalue with multiplicity k. Finally, we determine the trees with a non-zero R-eigenvalue of maximal possible multiplicity.
2010 Mathematics Subject Classifications:
Disclosure statement
No potential conflict of interest was reported by the author.