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

For an incidence geometry $$\mathcal{G}$ = ($\mathcal{P}$, $\mathcal{L}$, $\text{I}$)$withalinearrepresentation$$\mathcal{T}$_n^*($\mathcal{K}$)$, we apply WQH switching to construct a non-geometric graph ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ cospectral with the line graph $\mathrm{\Gamma }$ of $\mathcal{G}$. As an application, we show that for $h\ge 2$ 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 $(s,t,\alpha) = ((2^h+1)(2^m-1), 2^h-1, 2^m-1)$.