Bounds for Different Spreads of Line and Total Graphs

E. Andrade; E. Lenes; E. Mallea; M. Robbiano; Jonnathan Rodríguez

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arXiv · 1807.02899

Bounds for Different Spreads of Line and Total Graphs

E. Andrade·E. Lenes·E. Mallea·M. Robbiano·Jonnathan Rodríguez

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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.

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BibTeX RIS

E. Andrade, E. Lenes, E. Mallea, M. Robbiano, Jonnathan Rodríguez. 2018-07-08. Bounds for Different Spreads of Line and Total Graphs. https://arxiv.org/abs/1807.02899

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