On distance Laplacian spread and Wiener index of a graph

TL;DR

This paper investigates the distance Laplacian spread of graphs, establishing bounds related to the Wiener index, graph order, and maximum transmission degree, and characterizes extremal graphs, especially for k-partite graphs.

Contribution

It provides new bounds for the distance Laplacian spread in terms of graph invariants and characterizes extremal graphs, including specific results for k-partite graphs with disconnected complements.

Findings

Bounds for distance Laplacian spread in terms of Wiener index, order, and maximum transmission degree.

Lower bounds for k-partite graphs with disconnected complements, with equality conditions.

Characterization of extremal graphs achieving bounds, including complete k-partite graphs.

Abstract

Let $G$ be a simple connected simple graph of order $n$. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G), \partial_{2}^{L}(G),\dots,\partial_{n}^{L}(G)$. The \textit{ distance Laplacian spread} $DLS(G)$ of a connected graph $G$ is the difference between largest and second smallest distance Laplacian eigenvalues, that is, $\partial_{1}^{L}(G)-\partial_{n-1}^{L}(G)$. We obtain bounds for $DLS(G)$ in terms of the Wiener index $W(G)$, order $n$ and the maximum transmission degree $Tr_{max}(G)$ of $G$ and characterize the extremal graphs. We obtain two lower bounds for $DLS(G)$, the first one in terms of the order, diameter and the Wiener index of the graph, and the second one in terms of the order, maximum degree and the independence number of the graph. For a connected $ k-partite$ graph $G$, $k\leq n-1$, with $n$ vertices having disconnected complement, we show that $ DLS(G)\geq \Big \lfloor \frac{n}{k}\Big \rfloor$ with equality if and only if $G$ is a $complete ~ k-partite$ graph having cardinality of each independent class same and $n \equiv 0 \pmod k$.