Strongly Regular Graphs as Laplacian Extremal Graphs

TL;DR

This paper investigates the extremal properties of strongly regular graphs, showing they maximize Laplacian spread among graphs with fixed parameters, revealing their unique spectral extremality.

Contribution

It establishes that strongly regular graphs attain extremal Laplacian eigenvalues and spreads under various fixed graph parameters, identifying their spectral extremality.

Findings

Strongly regular graphs maximize Laplacian spread among fixed-parameter graphs.

They attain the maximum largest Laplacian eigenvalue.

They attain the minimum second-smallest Laplacian eigenvalue.

Abstract

The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also provided.