Bounds for Different Spreads of Line and Total Graphs

TL;DR

This paper investigates bounds on the spectral spread of line and total graphs, establishing inequalities and conditions relating to regularity, bipartiteness, and connectivity, with implications for spectral graph theory.

Contribution

It provides new bounds and conditions for the spread of line and total graphs, including characterizations for equality and bounds based on connectivity.

Findings

Spread of G is less than or equal to spread of its line graph for certain graphs.

Bounds for the spread of total graphs are derived based on regularity and connectivity.

Conditions for equality in spread inequalities are characterized.

Abstract

In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an $(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than or equal to the spread of its line graph, where the equality holds if and only if $G$ is regular bipartite. A sufficient condition for the spread of the graph not be greater than the signless Laplacian spread for a class of non bipartite and non regular graphs is proved. Additionally, we derive an upper bound for the spread of the line graph of graphs on $n$ vertices having a vertex (edge) connectivity less than or equal to a positive integer $k$. Combining techniques of interlacing of eigenvalues, we derive lower bounds for the Laplacian and signless Laplacian spread of the total graph of a connected graph. Moreover, for a regular graph, an upper and lower bound for the spread of its total graph is given.