An application of Hoffman graphs for spectral characterizations of graphs

TL;DR

This paper introduces the novel use of Hoffman graphs to spectral characterize certain classes of graphs, demonstrating that specific grid extensions are uniquely determined by their spectra for large parameters.

Contribution

It applies Hoffman graphs to spectral graph theory, establishing spectral characterization results for grid extensions and implications for distance-regular graphs.

Findings

2-clique extension of grid is spectrum-determined for large t

Grassmann graph J_2(2D,D) is determined by intersection numbers for large D

First application of Hoffman graphs in spectral characterizations

Abstract

In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the $2$-clique extension of the $(t+1)\times(t+1)$-grid is determined by its spectrum when $t$ is large enough. This result will help to show that the Grassmann graph $J_2(2D,D)$ is determined by its intersection numbers as a distance regular graph, if $D$ is large enough.