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.