# Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum

## TL;DR

This paper investigates when complete multipartite graphs are uniquely identified by their Seidel spectrum up to switching, showing that certain conditions guarantee this spectral determination, especially with multiple partition sets of the same size.

## Contribution

It establishes that complete multipartite graphs with multiple partition sets of the same size are determined by their Seidel spectrum up to switching, extending spectral characterization results.

## Key findings

- Graphs with the same spectrum are switching equivalent to complete multipartite graphs.
- Complete tripartite graphs with more than 18 vertices are conjectured to be determined by their Seidel spectrum.
- Conditions for spectral determination depend on the sizes and number of partition sets.

## Abstract

It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by Seidel switching has the same Seidel spectrum. We consider $G$ to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to $G$. It is shown that any graph which has the same spectrum as a complete $k$-partite graph is switching equivalent to a complete $k$-partite graph, and if the different partition sets sizes are $p_1,\ldots, p_l$, and there are at least three partition sets of each size $p_i$, $i=1,\ldots, l$, then $G$ is determined, up to switching, by its Seidel spectrum. Sufficient conditions for a complete tripartite graph to be determined by its Seidel spectrum are discussed, and a conjecture is made on complete tripartite graphs on more than 18 vertices.