A spectral characterization of the s-clique extension of the square grid graphs

TL;DR

This paper proves that for sufficiently large grid size, co-edge-regular graphs cospectral with the s-clique extension of a t-by-t grid are uniquely characterized as such extensions, using spectral graph theory techniques.

Contribution

It establishes a spectral characterization of the s-clique extension of grid graphs, confirming their uniqueness among co-edge-regular graphs for large parameters.

Findings

Co-edge-regular graphs cospectral with the s-clique extension are uniquely identified as such for large t.

Spectral methods can distinguish these grid extensions from other graphs.

Results extend to applications in characterizing distance-regular graphs like Grassmann graphs.

Abstract

In this paper we show that for integers $s\geq2$, $t\geq1$, any co-edge-regular graph which is cospectral with the $s$-clique extension of the $t\times t$-grid is the $s$-clique extension of the $t\times t$-grid, if $t$ is large enough. Gavrilyuk and Koolen used a weaker version of this result to show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection array as a distance-regular graph, if $D$ is large enough.