On the distance spectrum of generalized balanced trees

Abstract

For positive integers m 1 , m 2 , … , m h , a generalized balanced tree T ( m 1 , m 2 , … , m h ) is a rooted tree of height h such that every vertex of depth i has m i + 1 children, 0 ≤ i ≤ h − 1 . The distance matrix D ( G ) of a simple connected graph G of order n is an n × n matrix whose ( i , j ) th entry is the distance between i th and j th vertices. A connected graph G is called a k-partitioned transmission regular graph if there exists a vertex partition { V 1 , V 2 , … , V k } of G so that for 1 ≤ i , j ≤ k , and x ∈ V i , ∑ y ∈ V j d ( x , y ) is a constant. Here we show that T ( m 1 , m 2 , … , m h ) is an ( h + 1 ) -partitioned transmission regular graph. We find an ( h + 1 ) × ( h + 1 ) matrix whose largest eigenvalue is the distance spectral radius of T ( m 1 , m 2 , … , m h ) . We obtain the characteristic polynomial of D ( T ( m 1 , m 2 , … , m h ) ) in terms of that of the smaller matrices and give an idea to find the full spectrum. Moreover, we get that D ( T ( m 1 , m 2 , … , m h ) ) has − 2 an eigenvalue with multiplicity at least m 1 ⋯ m h − 1 ( m h − 1 ) and − ( m h + 2 ) ± m h ( m h + 4 ) as eigenvalues with multiplicity at least m 1 ⋯ m h − 2 ( m h − 1 − 1 ) .