Non-Geometric Cospectral Mates of Line Graphs with a Linear Representation

TL;DR

This paper constructs non-geometric cospectral graphs of line graphs using WQH switching and demonstrates the existence of certain strongly regular graphs that are not point graphs of partial geometries.

Contribution

It introduces a novel application of WQH switching to produce non-geometric cospectral graphs of line graphs, expanding understanding of graph spectra and geometric properties.

Findings

Constructed non-geometric cospectral graphs of line graphs.

Identified strongly regular graphs not arising from partial geometries.

Provided explicit parameters for these graphs.

Abstract

For an incidence geometry $\mathcal{G} = (\mathcal{P}, \mathcal{L}, \text{I})$ with a linear representation $\mathcal{T}_n^*(\mathcal{K})$, we apply WQH switching to construct a non-geometric graph $\Gamma'$ cospectral with the line graph $\Gamma$ of $\mathcal{G}$. As an application, we show that for $h \geq 2$ and $0 < m < h$, there are strongly regular graphs with parameters $(v, k, \lambda, \mu) = (2^{2h} (2^{m+h}+2^m-2^h), 2^h (2^h+1)(2^m-1), 2^h (2^{m+1}-3), 2^h (2^m-1))$ which are not point graphs of partial geometries of order $(s,t,\alpha) = ((2^h+1)(2^m-1), 2^h-1, 2^m-1)$.