On distance Laplacian spectral invariants of brooms and their complements

For a connected graph $G$ of order $n$, the distance Laplacian matrix $D^L(G)$ is defined as $D^L(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The largest eigenvalue of $D^L(G)$ is the distance Laplacian spectral radius of $G$ and the quantity $DLE(G)=\sum\limits_{i=1}^{n}|\rho^L_i(G)-\frac{2W(G)}{n}|$, where $W(G)$ is the Wiener index of $G$, is the distance Laplacian energy of $G$. Brooms of diameter $4$ are the trees obtained from the path $P_{5}$ by appending pendent vertices at some vertex of $ P_{5}$. One of the interesting and important problems in spectral graph theory is to find extremal graphs for a spectral graph invariant and ordering them according to this graph invariant. This problem has been considered for many families of graphs with respect to different graph matrices. In the present article, we consider this problem for brooms of diameter $4$ and their complements with respect to their distance Laplacian matrix. Formally, we discuss the distance Laplacian spectrum and the distance Laplacian energy of brooms of diameter $4$. We will prove that these families of trees can be ordered in terms of their distance Laplacian energy and the distance Laplacian spectral radius. Further, we obtain the distance Laplacian spectrum and the distance Laplacian energy of complement of the family of double brooms and order them in terms of the smallest non-zero distance Laplacian eigenvalue and the distance Laplacian energy.