### On the distance spectrum of trees

#### Abstract

Let $G$ be a connected graph with vertex set

$V(G)=\{v_{1},v_{2},\ldots,v_{n}\}$ and edge set $E(G).$ $D(G)=(d_{ij})_{n\times n}$ is the distance matrix

of $G,$ where $d_{ij}$ denotes the distance between $v_{i}$ and $v_{j}$. Let

$\lambda_{1}(D)\geq\lambda_{2}(D)\geq\cdots\geq\lambda_{n}(D)$ be

the distance spectrum of $G$. A graph $G$ is said to be determined

by its distance spectrum if any graph having the same distance spectrum as $G$ is isomorphic to $G$.

Trees can not be determined by its distance spectrum.

Naturally, we prove that two kinds of special trees path $P_{n}$ and double star $S(a,b)$ are determined by their distance spectra in this paper.

$V(G)=\{v_{1},v_{2},\ldots,v_{n}\}$ and edge set $E(G).$ $D(G)=(d_{ij})_{n\times n}$ is the distance matrix

of $G,$ where $d_{ij}$ denotes the distance between $v_{i}$ and $v_{j}$. Let

$\lambda_{1}(D)\geq\lambda_{2}(D)\geq\cdots\geq\lambda_{n}(D)$ be

the distance spectrum of $G$. A graph $G$ is said to be determined

by its distance spectrum if any graph having the same distance spectrum as $G$ is isomorphic to $G$.

Trees can not be determined by its distance spectrum.

Naturally, we prove that two kinds of special trees path $P_{n}$ and double star $S(a,b)$ are determined by their distance spectra in this paper.

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