On the distance spectrum of generalized balanced trees

Abstract

For positive integers $m_1,m_2,\dots ,m_h$, a generalized balanced tree $T(m_1,m_2,\dots ,m_h)$ is a rooted tree of height h such that every vertex of depth i has $m_{i+1}$ children, $0\le i\le h-1$. The distance matrix $D\left(G\right)$ of a simple connected graph G of order n is an $n\times n$ matrix whose $(i,j)\mathrm{th}$ entry is the distance between $i\mathrm{th}$ and $j\mathrm{th}$ vertices. A connected graph G is called a k-partitioned transmission regular graph if there exists a vertex partition $\{V_1,V_2,\dots ,V_k\}$ of G so that for $1\le i,j\le k$, and $x\in V_i$, $\sum _{y\in V_j}d(x,y)$ is a constant. Here we show that $T(m_1,m_2,\dots ,m_h)$ is an $(h+1)$-partitioned transmission regular graph. We find an $(h+1)\times (h+1)$ matrix whose largest eigenvalue is the distance spectral radius of $T(m_1,m_2,\dots ,m_h)$. We obtain the characteristic polynomial of $D\left(T\right(m_1,m_2,\dots ,m_h\left)\right)$ in terms of that of the smaller matrices and give an idea to find the full spectrum. Moreover, we get that $D\left(T\right(m_1,m_2,\dots ,m_h\left)\right)$ has $-2$ an eigenvalue with multiplicity at least $m_1\cdots m_{h-1}(m_h-1)$ and $-(m_h+2)\pm \sqrt{m_h(m_h+4)}$ as eigenvalues with multiplicity at least $m_1\cdots m_{h-2}(m_{h-1}-1)$.

KEYWORDS:

• Distance matrix
• distance spectrum
• distance characteristic polynomial
• quotient matrix
• generalized balanced trees

AMS CLASSIFICATIONS:

• 05C12
• 05C50

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).

Additional information

Funding

The first author is grateful to Council of Scientific and Industrial Research (CSIR), India [Grant number: 09/081(1283)/2016-EMR-I], for funding the research.