On the multiple eigenvalue of Randić matrix of trees

ABSTRACT

The Randić matrix (short for R-matrix) of a graph G is a symmetric matrix $R=\left(r_{ij}\right)$ whose $(i,j)$-entry is equal to $1/\sqrt{d_i{d}_j}$ if $v_i\sim v_j$, 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 $\rho =\frac{1}{\sqrt{2}}$ whenever $k>n/3$. Second, a tree T has at least k+1 pendant edges form an induced matching if it has $\frac{1}{\sqrt{2}}$ as a R-eigenvalue with multiplicity k. Finally, we determine the trees with a non-zero R-eigenvalue of maximal possible multiplicity.

Keywords:

• Randić matrix
• R-eigenvalues
• star complements
• tree

2010 Mathematics Subject Classifications:

• 05C50 Graphs and linear algebra (matrices
• eigenvalues
• etc.)

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Supported by the National Natural Science Foundation of China [grant numbers 11671344, 11531011, 11626205].