Abstract
For positive integers , a generalized balanced tree is a rooted tree of height h such that every vertex of depth i has children, . The distance matrix of a simple connected graph G of order n is an matrix whose entry is the distance between and vertices. A connected graph G is called a k-partitioned transmission regular graph if there exists a vertex partition of G so that for , and , is a constant. Here we show that is an -partitioned transmission regular graph. We find an matrix whose largest eigenvalue is the distance spectral radius of . We obtain the characteristic polynomial of in terms of that of the smaller matrices and give an idea to find the full spectrum. Moreover, we get that has an eigenvalue with multiplicity at least and as eigenvalues with multiplicity at least .
Acknowledgments
The authors are thankful to the referee for his/her valuable comments and suggestions which improved the presentation of the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).