The graphs of pyramids are determined by their spectrum

TL;DR

This paper proves that graphs of pyramids, a family of graphs formed by joining a complete graph with an independent set, are uniquely identified by their spectra, and explores spectral properties of stars and related graphs.

Contribution

It establishes that graphs of pyramids are determined by their spectra and characterizes when stars are spectrally determined, also analyzing positive definiteness of these graphs.

Findings

Graphs of pyramids are uniquely determined by their spectra.

Stars are spectrally determined iff their order is prime.

Graphs T_{n,k} are completely positive iff k ≤ 2.

Abstract

For natural numbers $k<n$ we study the graphs $T_{n,k}:=K_{k}\lor\overline{K_{n-k}}$. For $k=1$, $T_{n,1}$ is the star $S_{n-1}$. For $k>1$ we refer to $T_{n,k}$ as a \emph{graph of pyramids}. We prove that the graphs of pyramids are determined by their spectrum, and that a star $S_{n}$ is determined by its spectrum iff $n$ is prime. We also show that the graphs $T_{n,k}$ are completely positive iff $k\le2$.