# On the eigenvalues of some signed graphs

## TL;DR

This paper investigates the eigenvalues of the Seidel matrix for complete multipartite graphs, showing that under certain conditions, the Seidel spectrum uniquely determines the graph up to switching.

## Contribution

It provides a characterization of the Seidel eigenvalues for complete multipartite graphs and proves their spectral uniqueness under specific conditions.

## Key findings

- Seidel eigenvalues of complete multipartite graphs are studied.
- The Seidel spectrum determines the graph up to switching if at least three parts have size n_i.
- The Seidel characteristic polynomial is explicitly investigated.

## Abstract

Let $G$ be a simple graph and $A(G)$ be the adjacency matrix of $G$. The matrix $S(G) = J -I -2A(G)$ is called the Seidel matrix of $G$, where $I$ is an identity matrix and $J$ is a square matrix all of whose entries are equal to 1. Clearly, if $G$ is a graph of order $n$ with no isolated vertex, then the Seidel matrix of $G$ is also the adjacency matrix of a signed complete graph $K_n$ whose negative edges induce $G$. In this paper, we study the Seidel eigenvalues of the complete multipartite graph $K_{n_1,\ldots,n_k}$ and investigate its Seidel characteristic polynomial. We show that if there are at least three parts of size $n_i$, for some $i=1,\ldots,k$, then $K_{n_1,\ldots,n_k}$ is determined, up to switching, by its Seidel spectrum.