### Distance integral complete r-partite graphs

#### Abstract

Let $D(G)=(d_{ij})_{n\times n}$ denote the distance matrix of a

connected graph $G$ with order $n$, where $d_{ij}$ is equal to the

distance between vertices $v_{i}$ and $v_{j}$ in $G$. A graph is

called distance integral if all eigenvalues of its distance matrix

are integers. In this paper, we investigate distance integral

complete $r$-partite graphs

$K_{p_{1},p_{2},\ldots,p_{r}}=K_{a_{1}\cdot p_{1},a_{2}\cdot

p_{2},\ldots,a_{s}\cdot p_{s}}$ and give a sufficient and necessary

condition for $K_{a_{1}\cdot p_{1},a_{2}\cdot

p_{2},\ldots,a_{s}\cdot p_{s}}$ to be distance integral, from which

we construct infinitely many new classes of distance integral graphs

with $s=1,2,3,4$. Finally, we propose two basic open problems for

further study.

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