The graphs with all but two eigenvalues equal to $2$ or $-1$

TL;DR

This paper classifies graphs whose adjacency matrices have all but two eigenvalues equal to 2 or -1, identifying a specific class of generalized friendship graphs and determining which are uniquely identified by their spectra.

Contribution

The paper characterizes a class of graphs with specific spectral properties and identifies which among them are uniquely determined by their spectra.

Findings

Classified graphs with all but two eigenvalues as 2 or -1.

Identified the generalized friendship graphs $F_{t,r,k}$ with $t-r=3$.

Determined which graphs in this class are uniquely spectrum-determined.

Abstract

In this paper, all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $2$ and $-1$ are determined. These graphs conclude a class of generalized friendship graphs $F_{t,r,k}, $ which is the graph of $k$ copies of the complete graph $K_t$ meeting in common $r$ vertices such that $t-r=3.$ Which of these graphs are determined by its spectrum is are also obtained.