On distance integral and distance Laplacian integral graphs

TL;DR

This paper investigates conditions under which certain classes of graphs are distance integral or distance Laplacian integral, focusing on specific graph constructions and their eigenvalue properties.

Contribution

It provides new criteria for the distance and distance Laplacian integrality of complex graph families like joins and dumbbell graphs.

Findings

Conditions for $aar{K}_m abla C_n$ to be distance integral.

Conditions for $K_{p,p} abla C_n$ to be distance integral.

Criteria for $oldsymbol{DB}(W_{m,n})$ to be $D^L$-integral.

Abstract

Let $G$ be a connected graph on $n$ vertices and let $D(G)$ and $D^{L}(G)$ be the distance and the distance Laplacian matrices associated with $G$. A graph $G$ is said to be $D$-integral (resp. $D^L$-integral) if all eigenvalues of $D(G)$ (resp. $D^L(G)$) are integers. In this paper, we obtain various conditions under which the graphs $a\overline{K_m}\nabla C_n$ and $K_{p,p}\nabla C_n$ are distance integral. We also obtain conditions on $m$, $n$ under which the dumbbell graph $\boldsymbol{DB}(W_{m,n})$ is $D^L$-integral.